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Publications of Richard T. Durrett    :chronological  alphabetical  combined listing:

%% Books   
@book{fds299418,
   Author = {R.T. Durrett},
   Title = {Branching process models of cancer},
   Publisher = {Springer},
   Year = {2015},
   Key = {fds299418}
}

@book{fds177897,
   Author = {R. Durrett},
   Title = {Probability: Theory and Examples},
   Series = {4th Edition},
   Publisher = {Cambridge U Press},
   Year = {2010},
   Key = {fds177897}
}

@book{fds177879,
   Author = {R. Durrett},
   Title = {Elementary Probability for Applications},
   Series = {4th Edition},
   Pages = {x+243},
   Publisher = {Cambridge University Press},
   Address = {Cambridge},
   Year = {2009},
   ISBN = {978-0-521-86756-6},
   MRCLASS = {60-01 (60J10)},
   MRNUMBER = {MR2537423 (2010i:60003)},
   url = {http://www.ams.org/mathscinet-getitem?mr=2537423},
   Key = {fds177879}
}

@book{fds177878,
   Author = {R. Durrett},
   Title = {Probability Models of DNA Sequence Evolution},
   Series = {2nd Edition},
   Publisher = {Springer},
   Year = {2006},
   Key = {fds177878}
}

@book{fds177911,
   Author = {R. Durrett},
   Title = {Essentials of Stochastic Processes},
   Publisher = {Springer-Verlag},
   Year = {1998},
   Key = {fds177911}
}

@book{fds177910,
   Author = {R. Durrett},
   Title = {Stochastic Calculus: A Practrical Introduction},
   Publisher = {CRC Press},
   Year = {1996},
   Key = {fds177910}
}

@book{fds177912,
   Author = {R. Durrett},
   Title = {Lecture Notes on Particle Systems and Percolation},
   Publisher = {Wadsworth},
   Year = {1988},
   Key = {fds177912}
}

@book{fds177913,
   Author = {R. Durrett},
   Title = {Brownian Motion and Martingales in Analysis},
   Publisher = {Wadsworth},
   Year = {1984},
   Key = {fds177913}
}


%% Papers Published   
@article{fds341499,
   Author = {Cristali, I and Junge, M and Durrett, R},
   Title = {Poisson percolation on the oriented square
             lattice},
   Journal = {Stochastic Processes and Their Applications},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1016/j.spa.2019.01.005},
   Abstract = {© 2019 Elsevier B.V. In Poisson percolation each edge
             becomes open after an independent exponentially distributed
             time with rate that decreases in the distance from the
             origin. As a sequel to our work on the square lattice, we
             describe the limiting shape of the component containing the
             origin in the oriented case. We show that the density of
             occupied sites at height y in the cluster is close to the
             percolation probability in the corresponding homogeneous
             percolation process, and we study the fluctuations of the
             boundary.},
   Doi = {10.1016/j.spa.2019.01.005},
   Key = {fds341499}
}

@article{fds335535,
   Author = {Wang, Z and Durrett, R},
   Title = {Extrapolating weak selection in evolutionary
             games.},
   Journal = {Journal of Mathematical Biology},
   Volume = {78},
   Number = {1-2},
   Pages = {135-154},
   Year = {2019},
   Month = {January},
   url = {http://dx.doi.org/10.1007/s00285-018-1270-6},
   Abstract = {This work is inspired by a 2013 paper from Arne Traulsen's
             lab at the Max Plank Institute for Evolutionary Biology (Wu
             et al. in PLoS Comput Biol 9:e1003381, 2013). They studied
             evolutionary games when the mutation rate is so small that
             each mutation goes to fixation before the next one occurs.
             It has been shown that for [Formula: see text] games the
             ranking of the strategies does not change as strength of
             selection is increased (Wu et al. in Phys Rev 82:046106,
             2010). The point of the 2013 paper is that when there are
             three or more strategies the ordering can change as
             selection is increased. Wu et al. (2013) did numerical
             computations for a fixed population size N. Here, we will
             instead let the strength of selection [Formula: see text]
             where c is fixed and let [Formula: see text] to obtain
             formulas for the invadability probabilities [Formula: see
             text] that determine the rankings. These formulas, which are
             integrals on [0, 1], are intractable calculus problems, but
             can be easily evaluated numerically. Here, we use them to
             derive simple formulas for the ranking order when c is small
             or c is large.},
   Doi = {10.1007/s00285-018-1270-6},
   Key = {fds335535}
}

@article{fds337720,
   Author = {Ma, R and Durrett, R},
   Title = {A simple evolutionary game arising from the study of the
             role of igf-II in pancreatic cancer},
   Journal = {The Annals of Applied Probability},
   Volume = {28},
   Number = {5},
   Pages = {2896-2921},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2018},
   Month = {October},
   url = {http://dx.doi.org/10.1214/17-AAP1378},
   Abstract = {© Institute of Mathematical Statistics, 2018. We study an
             evolutionary game in which a producer at x gives birth at
             rate 1 to an offspring sent to a randomly chosen point in x
             + Nc, while a cheater at x gives birth at rate λ > 1 times
             the fraction of producers in x + Nd and sends its offspring
             to a randomly chosen point in x + Nc. We first study this
             game on the d-dimensional torus (Z mod L)d with Nd = (Z mod
             L)d and Nc = the 2d nearest neighbors. If we let L → ∞
             then t → ∞ the fraction of producers converges to 1/λ.
             In d ≥ 3 the limiting finite dimensional distributions
             converge as t → ∞ to the voter model equilibrium with
             density 1/λ. We next reformulate the system as an
             evolutionary game with “birth-death” updating and take
             Nc = Nd = N. Using results for voter model perturbations we
             show that in d = 3 with N = the six nearest neighbors, the
             density of producers converges to (2/λ) − 0.5 for 4/3 <
             λ < 4. Producers take over the system when λ < 4/3 and die
             out when λ > 4. In d = 2 with N = [−clog N, clog N]2
             there are similar phase transitions, with coexistence
             occurring when (1 + 2θ)/(1 + θ) < λ < (1 + 2θ)/θ where
             θ = (e3/(πc2) − 1)/2.},
   Doi = {10.1214/17-AAP1378},
   Key = {fds337720}
}

@article{fds339723,
   Author = {Talkington, A and Dantoin, C and Durrett, R},
   Title = {Ordinary Differential Equation Models for Adoptive
             Immunotherapy.},
   Journal = {Bulletin of Mathematical Biology},
   Volume = {80},
   Number = {5},
   Pages = {1059-1083},
   Year = {2018},
   Month = {May},
   url = {http://dx.doi.org/10.1007/s11538-017-0263-8},
   Abstract = {Modified T cells that have been engineered to recognize the
             CD19 surface marker have recently been shown to be very
             successful at treating acute lymphocytic leukemias. Here, we
             explore four previous approaches that have used ordinary
             differential equations to model this type of therapy,
             compare their properties, and modify the models to address
             their deficiencies. Although the four models treat the
             workings of the immune system in slightly different ways,
             they all predict that adoptive immunotherapy can be
             successful to move a patient from the large tumor fixed
             point to an equilibrium with little or no
             tumor.},
   Doi = {10.1007/s11538-017-0263-8},
   Key = {fds339723}
}

@article{fds330932,
   Author = {Huo, R and Durrett, R},
   Title = {Latent voter model on locally tree-like random
             graphs},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {128},
   Number = {5},
   Pages = {1590-1614},
   Publisher = {Elsevier BV},
   Year = {2018},
   Month = {May},
   url = {http://dx.doi.org/10.1016/j.spa.2017.08.004},
   Abstract = {© 2017 Elsevier B.V. In the latent voter model, individuals
             who have just changed their choice have a latent period,
             which is exponential with rate λ, during which they will
             not change their opinion. We study this model on random
             graphs generated by a configuration model with degrees
             3≤d(x)≤M. We show that if the number of vertices n→∞
             and logn≪λn≪n then there is a quasi-stationary state in
             which each opinion has probability ≈1∕2 and persists in
             this state for a time that is ≥nm for any
             m<∞.},
   Doi = {10.1016/j.spa.2017.08.004},
   Key = {fds330932}
}

@article{fds339577,
   Author = {Beckman, E and Dinan, E and Durrett, R and Huo, R and Junge,
             M},
   Title = {Asymptotic behavior of the brownian frog
             model},
   Journal = {Electronic Journal of Probability},
   Volume = {23},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1214/18-EJP215},
   Abstract = {© 2018, University of Washington. All rights reserved. We
             introduce an extension of the frog model to Euclidean space
             and prove properties for the spread of active particles. Fix
             r>0 and place a particle at each point x of a unit intensity
             Poisson point process P⊆ℝd−B(0,r). Around each point
             in P, put a ball of radius r. A particle at the origin
             performs Brownian motion. When it hits the ball around x for
             some x ∈ P, new particles begin independent Brownian
             motions from the centers of the balls in the cluster
             containing x. Subsequent visits to the cluster do nothing.
             This waking process continues indefinitely. For r smaller
             than the critical threshold of continuum percolation, we
             show that the set of activated points in P approximates a
             linearly expanding ball. Moreover, in any fixed ball the set
             of active particles converges to a unit intensity Poisson
             point process.},
   Doi = {10.1214/18-EJP215},
   Key = {fds339577}
}

@article{fds339578,
   Author = {Basak, A and Durrett, R and Foxall, E},
   Title = {Diffusion limit for the partner model at the critical
             value},
   Journal = {Electronic Journal of Probability},
   Volume = {23},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1214/18-EJP229},
   Abstract = {© 2018, University of Washington. All rights reserved.
             population with random formation and dissolution of
             partnerships, and with disease transmission only occuring
             within partnerships. Foxall, Edwards, and van den Driessche
             [7] found the critical value and studied the subcritical and
             supercritical regimes. Recently Foxall [4] has shown that
             (if there are enough initial infecteds I0) the extinction
             time in the critical model is of order √N. Here we improve
             that result by proving the convergence of
             iN(t)=I(√Nt)/√N to a limiting diffusion. We do this by
             showing that within a short time, this four dimensional
             process collapses to two dimensions: the number of SI and II
             partnerships are constant multiples of the the number of
             infected singles. The other variable, the total number of
             singles, fluctuates around its equilibrium like an
             Ornstein-Uhlenbeck process of magnitude √N on the original
             time scale and averages out of the limit theorem for iN(t).
             As a by-product of our proof we show that if τN is the
             extinction time of iN(t) (on the √N time scale) then τN
             has a limit.},
   Doi = {10.1214/18-EJP229},
   Key = {fds339578}
}

@article{fds339329,
   Author = {Cristali, I and Ranjan, V and Steinberg, J and Beckman, E and Durrett,
             R and Junge, M and Nolen, J},
   Title = {Block size in geometric(P)-biased permutations},
   Journal = {Electronic Communications in Probability},
   Volume = {23},
   Pages = {1-10},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2018},
   Month = {January},
   url = {http://dx.doi.org/10.1214/18-ECP182},
   Abstract = {© 2018, University of Washington. All rights reserved. Fix
             a probability distribution p = (p1, p2, …) on the positive
             integers. The first block in a p-biased permutation can be
             visualized in terms of raindrops that land at each positive
             integer j with probability pj. It is the first point K so
             that all sites in [1, K] are wet and all sites in (K, ∞)
             are dry. For the geometric distribution pj = p(1 − p)j−1
             we show that p log K converges in probability to an explicit
             constant as p tends to 0. Additionally, we prove that if p
             has a stretch exponential distribution, then K is infinite
             with positive probability.},
   Doi = {10.1214/18-ECP182},
   Key = {fds339329}
}

@article{fds330931,
   Author = {Lopatkin, AJ and Meredith, HR and Srimani, JK and Pfeiffer, C and Durrett, R and You, L},
   Title = {Persistence and reversal of plasmid-mediated antibiotic
             resistance.},
   Journal = {Nature Communications},
   Volume = {8},
   Number = {1},
   Pages = {1689},
   Year = {2017},
   Month = {November},
   url = {http://dx.doi.org/10.1038/s41467-017-01532-1},
   Abstract = {In the absence of antibiotic-mediated selection, sensitive
             bacteria are expected to displace their resistant
             counterparts if resistance genes are costly. However, many
             resistance genes persist for long periods in the absence of
             antibiotics. Horizontal gene transfer (primarily
             conjugation) could explain this persistence, but it has been
             suggested that very high conjugation rates would be
             required. Here, we show that common conjugal plasmids, even
             when costly, are indeed transferred at sufficiently high
             rates to be maintained in the absence of antibiotics in
             Escherichia coli. The notion is applicable to nine plasmids
             from six major incompatibility groups and mixed populations
             carrying multiple plasmids. These results suggest that
             reducing antibiotic use alone is likely insufficient for
             reversing resistance. Therefore, combining conjugation
             inhibition and promoting plasmid loss would be an effective
             strategy to limit conjugation-assisted persistence of
             antibiotic resistance.},
   Doi = {10.1038/s41467-017-01532-1},
   Key = {fds330931}
}

@article{fds329932,
   Author = {Gleeson, JP and Durrett, R},
   Title = {Temporal profiles of avalanches on networks.},
   Journal = {Nature Communications},
   Volume = {8},
   Number = {1},
   Pages = {1227},
   Year = {2017},
   Month = {October},
   url = {http://dx.doi.org/10.1038/s41467-017-01212-0},
   Abstract = {An avalanche or cascade occurs when one event causes one or
             more subsequent events, which in turn may cause further
             events in a chain reaction. Avalanching dynamics are studied
             in many disciplines, with a recent focus on average
             avalanche shapes, i.e., the temporal profiles of avalanches
             of fixed duration. At the critical point of the dynamics,
             the rescaled average avalanche shapes for different
             durations collapse onto a single universal curve. We apply
             Markov branching process theory to derive an equation
             governing the average avalanche shape for cascade dynamics
             on networks. Analysis of the equation at criticality
             demonstrates that nonsymmetric average avalanche shapes (as
             observed in some experiments) occur for certain combinations
             of dynamics and network topology. We give examples using
             numerical simulations of models for information spreading,
             neural dynamics, and behavior adoption and we propose simple
             experimental tests to quantify whether cascading systems are
             in the critical state.},
   Doi = {10.1038/s41467-017-01212-0},
   Key = {fds329932}
}

@article{fds329933,
   Author = {Tomasetti, C and Durrett, R and Kimmel, M and Lambert, A and Parmigiani,
             G and Zauber, A and Vogelstein, B},
   Title = {Role of stem-cell divisions in cancer risk},
   Journal = {Nature},
   Volume = {548},
   Number = {7666},
   Pages = {E13-E14},
   Publisher = {Springer Nature},
   Year = {2017},
   Month = {August},
   url = {http://dx.doi.org/10.1038/nature23302},
   Doi = {10.1038/nature23302},
   Key = {fds329933}
}

@article{fds327001,
   Author = {Nanda, M and Durrett, R},
   Title = {Spatial evolutionary games with weak selection},
   Journal = {Proceedings of the National Academy of Sciences of the
             United States of America},
   Volume = {114},
   Number = {23},
   Pages = {6046-6051},
   Year = {2017},
   Month = {June},
   url = {http://dx.doi.org/10.1073/pnas.1620852114},
   Abstract = {Recently, a rigorous mathematical theory has been developed
             for spatial games with weak selection, i.e., when the payoff
             differences between strategies are small. The key to the
             analysis is that when space and time are suitably rescaled,
             the spatial model converges to the solution of a partial
             differential equation (PDE). This approach can be used to
             analyze all 2 × 2 games, but there are a number of 3 × 3
             games for which the behavior of the limiting PDE is not
             known. In this paper, we give rules for determining the
             behavior of a large class of 3 × 3 games and check their
             validity using simulation. In words, the effect of space is
             equivalent to making changes in the payoff matrix, and once
             this is done, the behavior of the spatial game can be
             predicted from the behavior of the replicator equation for
             the modified game. We say predicted here because in some
             cases the behavior of the spatial game is different from
             that of the replicator equation for the modified game. For
             example, if a rock-paper-scissors game has a replicator
             equation that spirals out to the boundary, space stabilizes
             the system and produces an equilibrium.},
   Doi = {10.1073/pnas.1620852114},
   Key = {fds327001}
}

@article{fds323833,
   Author = {Bessonov, M and Durrett, R},
   Title = {Phase transitions for a planar quadratic contact
             process},
   Journal = {Advances in Applied Mathematics},
   Volume = {87},
   Pages = {82-107},
   Publisher = {Elsevier BV},
   Year = {2017},
   Month = {June},
   url = {http://dx.doi.org/10.1016/j.aam.2017.01.002},
   Doi = {10.1016/j.aam.2017.01.002},
   Key = {fds323833}
}

@article{fds323651,
   Author = {Durrett, R and Fan, W-TL},
   Title = {Genealogies in expanding populations},
   Journal = {The Annals of Applied Probability},
   Volume = {26},
   Number = {6},
   Pages = {3456-3490},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2016},
   Month = {December},
   url = {http://dx.doi.org/10.1214/16-aap1181},
   Doi = {10.1214/16-aap1181},
   Key = {fds323651}
}

@article{fds323652,
   Author = {Cox, JT and Durrett, R},
   Title = {Evolutionary games on the torus with weak
             selection},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {126},
   Number = {8},
   Pages = {2388-2409},
   Publisher = {Elsevier BV},
   Year = {2016},
   Month = {August},
   url = {http://dx.doi.org/10.1016/j.spa.2016.02.004},
   Doi = {10.1016/j.spa.2016.02.004},
   Key = {fds323652}
}

@article{fds321819,
   Author = {Ryser, MD and Worni, M and Turner, EL and Marks, JR and Durrett, R and Hwang, ES},
   Title = {Outcomes of Active Surveillance for Ductal Carcinoma in
             Situ: A Computational Risk Analysis.},
   Journal = {Journal of the National Cancer Institute},
   Volume = {108},
   Number = {5},
   Year = {2016},
   Month = {May},
   url = {http://dx.doi.org/10.1093/jnci/djv372},
   Abstract = {Ductal carcinoma in situ (DCIS) is a noninvasive breast
             lesion with uncertain risk for invasive progression. Usual
             care (UC) for DCIS consists of treatment upon diagnosis,
             thus potentially overtreating patients with low propensity
             for progression. One strategy to reduce overtreatment is
             active surveillance (AS), whereby DCIS is treated only upon
             detection of invasive disease. Our goal was to perform a
             quantitative evaluation of outcomes following an AS strategy
             for DCIS.Age-stratified, 10-year disease-specific cumulative
             mortality (DSCM) for AS was calculated using a computational
             risk projection model based upon published estimates for
             natural history parameters, and Surveillance, Epidemiology,
             and End Results data for outcomes. AS projections were
             compared with the DSCM for patients who received UC. To
             quantify the propagation of parameter uncertainty, a 95%
             projection range (PR) was computed, and sensitivity analyses
             were performed.Under the assumption that AS cannot
             outperform UC, the projected median differences in 10-year
             DSCM between AS and UC when diagnosed at ages 40, 55, and 70
             years were 2.6% (PR = 1.4%-5.1%), 1.5% (PR = 0.5%-3.5%), and
             0.6% (PR = 0.0%-2.4), respectively. Corresponding median
             numbers of patients needed to treat to avert one breast
             cancer death were 38.3 (PR = 19.7-69.9), 67.3 (PR =
             28.7-211.4), and 157.2 (PR = 41.1-3872.8), respectively.
             Sensitivity analyses showed that the parameter with greatest
             impact on DSCM was the probability of understaging invasive
             cancer at diagnosis.AS could be a viable management strategy
             for carefully selected DCIS patients, particularly among
             older age groups and those with substantial competing
             mortality risks. The effectiveness of AS could be markedly
             improved by reducing the rate of understaging.},
   Doi = {10.1093/jnci/djv372},
   Key = {fds321819}
}

@article{fds243415,
   Author = {Durrett, R and Foo, J and Leder, K},
   Title = {Spatial Moran models, II: cancer initiation in spatially
             structured tissue.},
   Journal = {Journal of Mathematical Biology},
   Volume = {72},
   Number = {5},
   Pages = {1369-1400},
   Year = {2016},
   Month = {April},
   ISSN = {0303-6812},
   url = {http://dx.doi.org/10.1007/s00285-015-0912-1},
   Abstract = {We study the accumulation and spread of advantageous
             mutations in a spatial stochastic model of cancer initiation
             on a lattice. The parameters of this general model can be
             tuned to study a variety of cancer types and genetic
             progression pathways. This investigation contributes to an
             understanding of how the selective advantage of cancer cells
             together with the rates of mutations driving cancer, impact
             the process and timing of carcinogenesis. These results can
             be used to give insights into tumor heterogeneity and the
             "cancer field effect," the observation that a malignancy is
             often surrounded by cells that have undergone premalignant
             transformation.},
   Doi = {10.1007/s00285-015-0912-1},
   Key = {fds243415}
}

@article{fds243417,
   Author = {Durrett, R and Zhang, Y},
   Title = {Coexistence of grass, saplings and trees in the
             Staver–Levin forest model},
   Journal = {The Annals of Applied Probability},
   Volume = {25},
   Number = {6},
   Pages = {3434-3464},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2015},
   Month = {December},
   ISSN = {1050-5164},
   url = {http://dx.doi.org/10.1214/14-aap1079},
   Doi = {10.1214/14-aap1079},
   Key = {fds243417}
}

@article{fds302176,
   Author = {Talkington, A and Durrett, R},
   Title = {Estimating Tumor Growth Rates In Vivo.},
   Journal = {Bulletin of Mathematical Biology},
   Volume = {77},
   Number = {10},
   Pages = {1934-1954},
   Year = {2015},
   Month = {October},
   ISSN = {0092-8240},
   url = {http://dx.doi.org/10.1007/s11538-015-0110-8},
   Abstract = {In this paper, we develop methods for inferring tumor growth
             rates from the observation of tumor volumes at two time
             points. We fit power law, exponential, Gompertz, and
             Spratt’s generalized logistic model to five data sets.
             Though the data sets are small and there are biases due to
             the way the samples were ascertained, there is a clear sign
             of exponential growth for the breast and liver cancers, and
             a 2/3’s power law (surface growth) for the two
             neurological cancers.},
   Doi = {10.1007/s11538-015-0110-8},
   Key = {fds302176}
}

@article{fds323653,
   Author = {Varghese, C and Durrett, R},
   Title = {Spatial networks evolving to reduce length},
   Journal = {Journal of Complex Networks},
   Volume = {3},
   Number = {3},
   Pages = {411-430},
   Publisher = {Oxford University Press (OUP)},
   Year = {2015},
   Month = {September},
   url = {http://dx.doi.org/10.1093/comnet/cnu044},
   Doi = {10.1093/comnet/cnu044},
   Key = {fds323653}
}

@article{fds243418,
   Author = {Ryser, MD and Myers, ER and Durrett, R},
   Title = {HPV clearance and the neglected role of stochasticity.},
   Journal = {Plos Computational Biology},
   Volume = {11},
   Number = {3},
   Pages = {e1004113},
   Year = {2015},
   Month = {March},
   ISSN = {1553-734X},
   url = {http://hdl.handle.net/10161/9545 Duke open
             access},
   Abstract = {Clearance of anogenital and oropharyngeal HPV infections is
             attributed primarily to a successful adaptive immune
             response. To date, little attention has been paid to the
             potential role of stochastic cell dynamics in the time it
             takes to clear an HPV infection. In this study, we combine
             mechanistic mathematical models at the cellular level with
             epidemiological data at the population level to disentangle
             the respective roles of immune capacity and cell dynamics in
             the clearing mechanism. Our results suggest that chance-in
             form of the stochastic dynamics of basal stem cells-plays a
             critical role in the elimination of HPV-infected cell
             clones. In particular, we find that in immunocompetent
             adolescents with cervical HPV infections, the immune
             response may contribute less than 20% to virus clearance-the
             rest is taken care of by the stochastic proliferation
             dynamics in the basal layer. In HIV-negative individuals,
             the contribution of the immune response may be
             negligible.},
   Doi = {10.1371/journal.pcbi.1004113},
   Key = {fds243418}
}

@article{fds243416,
   Author = {Durrett, R and Moseley, S},
   Title = {Spatial Moran models I. Stochastic tunneling in the neutral
             case},
   Journal = {The Annals of Applied Probability},
   Volume = {25},
   Number = {1},
   Pages = {104-115},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2015},
   Month = {February},
   ISSN = {1050-5164},
   url = {http://dx.doi.org/10.1214/13-aap989},
   Doi = {10.1214/13-aap989},
   Key = {fds243416}
}

@article{fds323654,
   Author = {Magura, SR and Pong, VH and Durrett, R and Sivakoff,
             D},
   Title = {Two evolving social network models},
   Journal = {Alea},
   Volume = {12},
   Number = {2},
   Pages = {699-715},
   Year = {2015},
   Month = {January},
   Abstract = {In our first model, individuals have opinions in [0, 1] d .
             Connections are broken at rate proportional to their length
             ℓ, an end point is chosen at random, a new connection to a
             random individual is proposed. In version (i) the new edge
             is always accepted. In version (ii) a new connection of
             length ℓ' is accepted with probability minℓ/ℓ', 1. Our
             second model is a dynamic version of preferential
             attachment. Edges are chosen at random for deletion, then
             one endpoint chosen at random connects to vertex z with
             probability proportional to f(d(z)), where d(z) is the
             degree of z, f(k) = θ(k+1)+(1-θ)(d+1), d is the average
             degree. In words, this is a mixture of degree-proportional,
             at random rewiring. The common feature of these models is
             that they have stationary distributions that satisfy the
             detailed balance condition, are given by explicit formulas.
             In addition, the equilibrium of the first model is closely
             related to long range percolation, of the second to the
             configuration model of random graphs. As a result, we obtain
             explicit results about the degree distribution,
             connectivity, diameter for each model.},
   Key = {fds323654}
}

@article{fds243413,
   Author = {Durrett, R},
   Title = {Spatial evolutionary games with small selection
             coefficients},
   Journal = {Electronic Journal of Probability},
   Volume = {19},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2014},
   Month = {December},
   url = {http://dx.doi.org/10.1214/EJP.v19-3621},
   Abstract = {© 2015 University of Washington. All rights reserved. Here
             we will use results of Cox, Durrett, and Perkins [56] for
             voter model perturbations to study spatial evolutionary
             games on ℤ<sup>d</sup>, d ≥ 3 when the interaction
             kernel is finite range, symmetric, and has covariance matrix
             σ<sup>2</sup>I. The games we consider have payoff matrices
             of the form 1 + wG where 1 is matrix of all 1’s and w is
             small and positive. Since our population size N = ∞, we
             call our selection small rather than weak which usually
             means w = O(1=N). The key to studying these games is the
             fact that when the dynamics are suitably rescaled in space
             and time they convergence to solutions of a reaction
             diffusion equation (RDE). Inspired by work of Ohtsuki and
             Nowak [28] and Tarnita et al [35, 36] we show that the
             reaction term is the replicator equation for a modified game
             matrix and the modifications of the game matrix depend on
             the interaction kernel only through the values of two or
             three simple probabilities for an associated coalescing
             random walk. Two strategy games lead to an RDE with a cubic
             nonlinearity, so we can describe the phase diagram
             completely. Three strategy games lead to a pair of coupled
             RDE, but using an idea from our earlier work [59], we are
             able to show that if there is a repelling function for the
             replicator equation for the modified game, then there is
             coexistence in the spatial game when selection is small.
             This enables us to prove coexistence in the spatial model in
             a wide variety of examples including the behavior of four
             evolutionary games that have recently been used in cancer
             modeling.},
   Doi = {10.1214/EJP.v19-3621},
   Key = {fds243413}
}

@article{fds243414,
   Author = {Aristotelous, AC and Durrett, R},
   Title = {Fingering in Stochastic Growth Models},
   Journal = {Experimental Mathematics},
   Volume = {23},
   Number = {4},
   Pages = {465-474},
   Publisher = {Informa UK Limited},
   Year = {2014},
   Month = {October},
   ISSN = {1058-6458},
   url = {http://dx.doi.org/10.1080/10586458.2014.947053},
   Doi = {10.1080/10586458.2014.947053},
   Key = {fds243414}
}

@article{fds243419,
   Author = {Durrett, R and Zhang, Y},
   Title = {Exact solution for a metapopulation version of Schelling's
             model.},
   Journal = {Proceedings of the National Academy of Sciences of the
             United States of America},
   Volume = {111},
   Number = {39},
   Pages = {14036-14041},
   Year = {2014},
   Month = {September},
   ISSN = {0027-8424},
   url = {http://dx.doi.org/10.1073/pnas.1414915111},
   Abstract = {In 1971, Schelling introduced a model in which families move
             if they have too many neighbors of the opposite type. In
             this paper, we will consider a metapopulation version of the
             model in which a city is divided into N neighborhoods, each
             of which has L houses. There are ρNL red families and ρNL
             blue families for some ρ < 1/2. Families are happy if there
             are ≤ ρ(c)L families of the opposite type in their
             neighborhood and unhappy otherwise. Each family moves to
             each vacant house at rates that depend on their happiness at
             their current location and that of their destination. Our
             main result is that if neighborhoods are large, then there
             are critical values ρ(b) < ρ(d) < ρ(c), so that for ρ <
             ρ(b), the two types are distributed randomly in
             equilibrium. When ρ > ρ(b), a new segregated equilibrium
             appears; for ρ(b) < ρ < ρ(d), there is bistability, but
             when ρ increases past ρ(d) the random state is no longer
             stable. When ρ(c) is small enough, the random state will
             again be the stationary distribution when ρ is close to
             1/2. If so, this is preceded by a region of
             bistability.},
   Doi = {10.1073/pnas.1414915111},
   Key = {fds243419}
}

@article{fds243421,
   Author = {Aristotelous, AC and Durrett, R},
   Title = {Chemical evolutionary games.},
   Journal = {Theoretical Population Biology},
   Volume = {93},
   Pages = {1-13},
   Year = {2014},
   Month = {May},
   ISSN = {0040-5809},
   url = {http://dx.doi.org/10.1016/j.tpb.2014.02.001},
   Abstract = {Inspired by the use of hybrid cellular automata in modeling
             cancer, we introduce a generalization of evolutionary games
             in which cells produce and absorb chemicals, and the
             chemical concentrations dictate the death rates of cells and
             their fitnesses. Our long term aim is to understand how the
             details of the interactions in a system with n species and m
             chemicals translate into the qualitative behavior of the
             system. Here, we study two simple 2×2 games with two
             chemicals and revisit the two and three species versions of
             the one chemical colicin system studied earlier by Durrett
             and Levin (1997). We find that in the 2×2 examples, the
             behavior of our new spatial model can be predicted from that
             of the mean field differential equation using ideas of
             Durrett and Levin (1994). However, in the three species
             colicin model, the system with diffusion does not have the
             coexistence which occurs in the lattices model in which
             sites interact with only their nearest neighbors.},
   Doi = {10.1016/j.tpb.2014.02.001},
   Key = {fds243421}
}

@article{fds243420,
   Author = {Durrett, R and Liggett, T and Zhang, Y},
   Title = {The contact process with fast voting},
   Journal = {Electronic Journal of Probability},
   Volume = {19},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2014},
   Month = {March},
   url = {http://dx.doi.org/10.1214/EJP.v19-3021},
   Abstract = {Consider a combination of the contact process and the voter
             model in which deaths occur at rate 1 per site, and across
             each edge between nearest neighbors births occur at rate λ
             and voting events occur at rate θ. We are interested in the
             asymptotics as θ→∞ of the critical value λc(θ) for
             the existence of a nontrivial stationary distribution. In
             d≥3, λc(θ)→1/(2dρd) where ρd is the probability a d
             dimensional simple random walk does not return to its
             starting point.In d=2, λc(θ)/log(θ)→1/4π, while in
             d=1, λc(θ)/θ1/2 has lim inf≥1/2√ and lim sup<∞.The
             lower bound might be the right answer, but proving this, or
             even getting a reasonable upper bound, seems to be a
             difficult problem.},
   Doi = {10.1214/EJP.v19-3021},
   Key = {fds243420}
}

@article{fds243422,
   Author = {Shi, F and Mucha, PJ and Durrett, R},
   Title = {Multiopinion coevolving voter model with infinitely many
             phase transitions},
   Journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter
             Physics},
   Volume = {88},
   Number = {6},
   Year = {2013},
   Month = {December},
   ISSN = {1539-3755},
   url = {http://dx.doi.org/10.1103/PhysRevE.88.062818},
   Doi = {10.1103/PhysRevE.88.062818},
   Key = {fds243422}
}

@article{fds243423,
   Author = {Varghese, C and Durrett, R},
   Title = {Phase transitions in the quadratic contact process on
             complex networks},
   Journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter
             Physics},
   Volume = {87},
   Number = {6},
   Pages = {paper 062819},
   Publisher = {American Physical Society (APS)},
   Year = {2013},
   Month = {June},
   ISSN = {1539-3755},
   url = {http://dx.doi.org/10.1103/PhysRevE.87.062819},
   Abstract = {The quadratic contact process (QCP) is a natural extension
             of the well-studied linear contact process where infected
             (1) individuals infect susceptible (0) neighbors at rate λ
             and infected individuals recover (1ï·0) at rate 1. In the
             QCP, a combination of two 1's is required to effect a 0ï·1
             change. We extend the study of the QCP, which so far has
             been limited to lattices, to complex networks. We define two
             versions of the QCP: vertex-centered (VQCP) and
             edge-centered (EQCP) with birth events 1-0-1ï·1-1-1 and
             1-1-0ï·1-1-1, respectively, where "-" represents an edge.
             We investigate the effects of network topology by
             considering the QCP on random regular, Erdos-Rényi, and
             power-law random graphs. We perform mean-field calculations
             as well as simulations to find the steady-state fraction of
             occupied vertices as a function of the birth rate. We find
             that on the random regular and Erdos-Rényi graphs, there is
             a discontinuous phase transition with a region of
             bistability, whereas on the heavy-tailed power-law graph,
             the transition is continuous. The critical birth rate is
             found to be positive in the former but zero in the latter.
             © 2013 American Physical Society.},
   Doi = {10.1103/PhysRevE.87.062819},
   Key = {fds243423}
}

@article{fds243424,
   Author = {Mode, CJ and Durrett, R and Klebaner, F and Olofsson,
             P},
   Title = {Applications of Stochastic Processes in Biology and
             Medicine},
   Journal = {International Journal of Stochastic Analysis},
   Volume = {2013},
   Pages = {1-2},
   Publisher = {Hindawi Limited},
   Year = {2013},
   Month = {March},
   ISSN = {2090-3332},
   url = {http://dx.doi.org/10.1155/2013/790625},
   Doi = {10.1155/2013/790625},
   Key = {fds243424}
}

@article{fds243425,
   Author = {Durrett, R},
   Title = {Cancer Modeling: A Personal Perspective},
   Journal = {Notices of the American Mathematical Society},
   Volume = {60},
   Number = {03},
   Pages = {304-304},
   Publisher = {American Mathematical Society (AMS)},
   Year = {2013},
   Month = {March},
   ISSN = {0002-9920},
   url = {http://dx.doi.org/10.1090/noti953},
   Doi = {10.1090/noti953},
   Key = {fds243425}
}

@article{fds243517,
   Author = {Chatterjee, S and Durrett, R},
   Title = {A first order phase transition in the threshold
             ??θ2
             contact process on random ??r-regular
             graphs and ??r-trees},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {123},
   Number = {2},
   Pages = {561-578},
   Publisher = {Elsevier BV},
   Year = {2013},
   Month = {February},
   url = {http://dx.doi.org/10.1016/j.spa.2012.10.001},
   Doi = {10.1016/j.spa.2012.10.001},
   Key = {fds243517}
}

@article{fds243526,
   Author = {Durrett, R},
   Title = {Population genetics of neutral mutations in exponentially
             growing cancer cell populations},
   Journal = {The Annals of Applied Probability},
   Volume = {23},
   Number = {1},
   Pages = {230-250},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2013},
   Month = {February},
   url = {http://dx.doi.org/10.1214/11-AAP824},
   Abstract = {In order to analyze data from cancer genome sequencing
             projects, we need to be able to distinguish causative, or
             "driver," mutations from "passenger" mutations that have no
             selective effect. Toward this end, we prove results
             concerning the frequency of neutural mutations in
             exponentially growing multitype branching processes that
             have been widely used in cancer modeling. Our results yield
             a simple new population genetics result for the site
             frequency spectrum of a sample from an exponentially growing
             population. © Institute of Mathematical Statistics,
             2013.},
   Doi = {10.1214/11-AAP824},
   Key = {fds243526}
}

@article{fds243516,
   Author = {Danesh, K and Durrett, R and Havrilesky, LJ and Myers,
             E},
   Title = {A branching process model of ovarian cancer.},
   Journal = {Journal of Theoretical Biology},
   Volume = {314},
   Pages = {10-15},
   Year = {2012},
   Month = {December},
   url = {http://www.ncbi.nlm.nih.gov/pubmed/22959913},
   Abstract = {Ovarian cancer is usually diagnosed at an advanced stage,
             rendering the possibility of cure unlikely. To date, no
             cost-effective screening test has proven effective for
             reducing mortality. To estimate the window of opportunity
             for ovarian cancer screening, we develop a branching process
             model for ovarian cancer growth and progression accounting
             for three cell populations: Primary (cells in the ovary or
             fallopian tube), Peritoneal (viable cells in peritoneal
             fluid), and Metastatic (cells implanted on other
             intra-abdominal surfaces). Growth and migration parameters
             were chosen to match results of clinical studies. Using
             these values, our model predicts a window of opportunity of
             2.9 years, indicating that one would have to screen at least
             every other year to be effective. The model can be used to
             inform future efforts in designing improved screening and
             treatment strategies.},
   Doi = {10.1016/j.jtbi.2012.08.025},
   Key = {fds243516}
}

@article{fds243519,
   Author = {Durrett, R and Gleeson, JP and Lloyd, AL and Mucha, PJ and Shi, F and Sivakoff, D and Socolar, JES and Varghese, C},
   Title = {Graph fission in an evolving voter model.},
   Journal = {Proc Natl Acad Sci U S A},
   Volume = {109},
   Number = {10},
   Pages = {3682-3687},
   Year = {2012},
   Month = {March},
   url = {http://www.ncbi.nlm.nih.gov/pubmed/22355142},
   Abstract = {We consider a simplified model of a social network in which
             individuals have one of two opinions (called 0 and 1) and
             their opinions and the network connections coevolve. Edges
             are picked at random. If the two connected individuals hold
             different opinions then, with probability 1 - α, one
             imitates the opinion of the other; otherwise (i.e., with
             probability α), the link between them is broken and one of
             them makes a new connection to an individual chosen at
             random (i) from those with the same opinion or (ii) from the
             network as a whole. The evolution of the system stops when
             there are no longer any discordant edges connecting
             individuals with different opinions. Letting ρ be the
             fraction of voters holding the minority opinion after the
             evolution stops, we are interested in how ρ depends on α
             and the initial fraction u of voters with opinion 1. In case
             (i), there is a critical value α(c) which does not depend
             on u, with ρ ≈ u for α > α(c) and ρ ≈ 0 for
             α < α(c). In case (ii), the transition point α(c)(u)
             depends on the initial density u. For α > α(c)(u),
             ρ ≈ u, but for α < α(c)(u), we have
             ρ(α,u) = ρ(α,1/2). Using simulations and approximate
             calculations, we explain why these two nearly identical
             models have such dramatically different phase
             transitions.},
   Doi = {10.1073/pnas.1200709109},
   Key = {fds243519}
}

@article{fds243518,
   Author = {Durrett, R and Remenik, D},
   Title = {Evolution of dispersal distance},
   Journal = {Journal of Mathematical Biology},
   Volume = {64},
   Number = {4},
   Pages = {657-666},
   Year = {2012},
   ISSN = {0303-6812},
   url = {http://dx.doi.org/10.1007/s00285-011-0444-2},
   Abstract = {The problem of how often to disperse in a randomly
             fluctuating environment has long been investigated,
             primarily using patch models with uniform dispersal. Here,
             we consider the problem of choice of seed size for plants in
             a stable environment when there is a trade off between
             survivability and dispersal range. Ezoe (J Theor Biol
             190:287-293, 1998) and Levin and Muller-Landau (Evol Ecol
             Res 2:409-435, 2000) approached this problem using models
             that were essentially deterministic, and used calculus to
             find optimal dispersal parameters. Here we follow Hiebeler
             (Theor Pop Biol 66:205-218, 2004) and use a stochastic
             spatial model to study the competition of different
             dispersal strategies. Most work on such systems is done by
             simulation or nonrigorous methods such as pair
             approximation. Here, we use machinery developed by Cox et
             al. (Voter model perturbations and reaction diffusion
             equations 2011) to rigorously and explicitly compute
             evolutionarily stable strategies. © 2011
             Springer-Verlag.},
   Doi = {10.1007/s00285-011-0444-2},
   Key = {fds243518}
}

@article{fds243520,
   Author = {Chatterjee, S and Durrett, R},
   Title = {Asymptotic behavior of Aldous’ gossip process},
   Journal = {The Annals of Applied Probability},
   Volume = {21},
   Number = {6},
   Pages = {2447-2482},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2011},
   Month = {December},
   ISSN = {1050-5164},
   url = {http://dx.doi.org/10.1214/10-aap750},
   Abstract = {Aldous [(2007) Preprint] defined a gossip process in which
             space is a discrete N × N torus, and the state of the
             process at time t is the set of individuals who know the
             information. Information spreads from a site to its nearest
             neighbors at rate 1/4 each and at rate N-α to a site chosen
             at random from the torus. We will be interested in the case
             in which α &lt; 3, where the long range transmission
             significantly accelerates the time at which everyone knows
             the information. We prove three results that precisely
             describe the spread of information in a slightly simplified
             model on the real torus. The time until everyone knows the
             information is asymptotically T = (2 - 2α/3)Nα/3 logN. If
             ρs is the fraction of the population who know the
             information at time s and ε is small then, for large N, the
             time until ρs reaches ε is T (ε) ~ T + Nα/3 log(3ε/M),
             where M is a random variable determined by the early spread
             of the information. The value of ρs at time s = T (1/3) +
             tNα/3 is almost a deterministic function h(t) which
             satisfies an odd looking integro-differential equation. The
             last result confirms a heuristic calculation of Aldous. ©
             Institute of Mathematical Statistics, 2011.},
   Doi = {10.1214/10-aap750},
   Key = {fds243520}
}

@article{fds243521,
   Author = {Durrett, R and Remenik, D},
   Title = {Brunet-derrida particle systems, free boundary problems and
             wiener-hopf equations},
   Journal = {The Annals of Probability},
   Volume = {39},
   Number = {6},
   Pages = {2043-2078},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2011},
   Month = {November},
   ISSN = {0091-1798},
   url = {http://dx.doi.org/10.1214/10-AOP601},
   Abstract = {We consider a branching-selection system in ℝ with N
             particles which give birth independently at rate 1 and where
             after each birth the leftmost particle is erased, keeping
             the number of particles constant. We show that, as N →∞,
             the empirical measure process associated to the system
             converges in distribution to a deterministic measure-valued
             process whose densities solve a free boundary
             integro-differential equation. We also show that this
             equation has a unique traveling wave solution traveling at
             speed c or no such solution depending on whether c ≥ a or
             c>a,wherea is the asymptotic speed of the branching random
             walk obtained by ignoring the removal of the leftmost
             particles in our process. The traveling wave solutions
             correspond to solutions of Wiener-Hopf equations. © 2011
             Institute of Mathematical Statistics.},
   Doi = {10.1214/10-AOP601},
   Key = {fds243521}
}

@article{fds243523,
   Author = {Chatterjee, S and Durrett, R},
   Title = {Persistence of activity in threshold contact processes, an
             “Annealed approximation” of random Boolean
             networks},
   Journal = {Random Structures & Algorithms},
   Volume = {39},
   Number = {2},
   Pages = {228-246},
   Publisher = {WILEY},
   Year = {2011},
   Month = {September},
   ISSN = {1042-9832},
   MRCLASS = {60K35 (05C80)},
   MRNUMBER = {2850270},
   url = {http://dx.doi.org/10.1002/rsa.20357},
   Abstract = {We consider a model for gene regulatory networks that is a
             modification of Kauffmann's J Theor Biol 22 (1969), 437-467
             random Boolean networks. There are three parameters: $n =
             {\rm the}$ number of nodes, $r = {\rm the}$ number of inputs
             to each node, and $p = {\rm the}$ expected fraction of 1'sin
             the Boolean functions at each node. Following a standard
             practice in thephysics literature, we use a threshold
             contact process on a random graph on n nodes, in which each
             node has in degree r, to approximate its dynamics. We show
             that if $r\ge 3$ and $r \cdot 2p(1-p)&gt;1$, then the
             threshold contact process persists for a long time, which
             correspond to chaotic behavior of the Boolean network.
             Unfortunately, we are only able to prove the persistence
             time is $\ge \exp(cn^{b(p)})$ with $b(p)&gt;0$ when $r\cdot
             2p(1-p)&gt; 1$, and $b(p)=1$ when $(r-1)\cdot 2p(1-p)&gt;1$.
             © 2011 Wiley Periodicals, Inc..},
   Doi = {10.1002/rsa.20357},
   Key = {fds243523}
}

@article{fds243529,
   Author = {Durrett, R and Foo, J and Leder, K and Mayberry, J and Michor,
             F},
   Title = {Intratumor heterogeneity in evolutionary models of tumor
             progression.},
   Journal = {Genetics},
   Volume = {188},
   Number = {2},
   Pages = {461-477},
   Year = {2011},
   Month = {June},
   ISSN = {0016-6731},
   url = {http://dx.doi.org/10.1534/genetics.110.125724},
   Abstract = {With rare exceptions, human tumors arise from single cells
             that have accumulated the necessary number and types of
             heritable alterations. Each such cell leads to dysregulated
             growth and eventually the formation of a tumor. Despite
             their monoclonal origin, at the time of diagnosis most
             tumors show a striking amount of intratumor heterogeneity in
             all measurable phenotypes; such heterogeneity has
             implications for diagnosis, treatment efficacy, and the
             identification of drug targets. An understanding of the
             extent and evolution of intratumor heterogeneity is
             therefore of direct clinical importance. In this article, we
             investigate the evolutionary dynamics of heterogeneity
             arising during exponential expansion of a tumor cell
             population, in which heritable alterations confer random
             fitness changes to cells. We obtain analytical estimates for
             the extent of heterogeneity and quantify the effects of
             system parameters on this tumor trait. Our work contributes
             to a mathematical understanding of intratumor heterogeneity
             and is also applicable to organisms like bacteria,
             agricultural pests, and other microbes.},
   Doi = {10.1534/genetics.110.125724},
   Key = {fds243529}
}

@article{fds243522,
   Author = {Durrett, R and Mayberry, J},
   Title = {Traveling waves of selective sweeps},
   Journal = {The Annals of Applied Probability},
   Volume = {21},
   Number = {2},
   Pages = {699-744},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2011},
   Month = {April},
   ISSN = {1050-5164},
   MRCLASS = {60J85 (92D25)},
   MRNUMBER = {2807971},
   url = {http://dx.doi.org/10.1214/10-AAP721},
   Abstract = {The goal of cancer genome sequencing projects is to
             determine the genetic alterations that cause common cancers.
             Many malignancies arise during the clonal expansion of a
             benign tumor which motivates the study of recurrent
             selective sweeps in an exponentially growing population. To
             better understand this process, Beerenwinkel et al. [PLoS
             Comput. Biol. 3 (2007) 2239- 2246] consider a Wright-Fisher
             model in which cells from an exponentially growing
             population accumulate advantageous mutations. Simulations
             show a traveling wave in which the time of the first k-fold
             mutant, Tk, is approximately linear in k and heuristics are
             used to obtain formulas for ETk. Here, we consider the
             analogous problem for the Moran model and prove that as the
             mutation rate μ →0, Tk ∼ ck log(1/μ), where the ck can
             be computed explicitly. In addition, we derive a limiting
             result on a log scale for the size of Xk(t) = the number of
             cells with k mutations at time t . © Institute of
             Mathematical Statistics, 2011.},
   Doi = {10.1214/10-AAP721},
   Key = {fds243522}
}

@article{fds243527,
   Author = {Cox, JT and Durrett, R and Perkins, E},
   Title = {Voter model perturbations and reaction diffusion
             equations},
   Journal = {Asterique},
   Volume = {349},
   Pages = {1-113},
   Year = {2011},
   url = {http://arxiv.org/abs/math/1103.1676},
   Key = {fds243527}
}

@article{fds243528,
   Author = {Durrett, R and Chatterjee, S},
   Title = {Persistence of activity in random Boolean
             networks},
   Journal = {Random Structures and Algorithms},
   Volume = {39},
   Pages = {228-246},
   Year = {2011},
   Key = {fds243528}
}

@article{fds243524,
   Author = {Durrett, R and Mayberry, J},
   Title = {Evolution in predator–prey systems},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {120},
   Number = {7},
   Pages = {1364-1392},
   Publisher = {Elsevier BV},
   Year = {2010},
   Month = {July},
   ISSN = {0304-4149},
   MRCLASS = {92D25 (60J60 60K35 92D15)},
   MRNUMBER = {2639750 (2011d:92087)},
   url = {http://dx.doi.org/10.1016/j.spa.2010.03.011},
   Abstract = {We study the adaptive dynamics of predatorprey systems
             modeled by a dynamical system in which the traits of
             predators and prey are allowed to evolve by small mutations.
             When only the prey are allowed to evolve, and the size of
             the mutational change tends to 0, the system does not
             exhibit long term prey coexistence and the trait of the
             resident prey type converges to the solution of an ODE. When
             only the predators are allowed to evolve, coexistence of
             predators occurs. In this case, depending on the parameters
             being varied, we see that (i) the number of coexisting
             predators remains tight and the differences in traits from a
             reference species converge in distribution to a limit, or
             (ii) the number of coexisting predators tends to infinity,
             and we calculate the asymptotic rate at which the traits of
             the least and most "fit" predators in the population
             increase. This last result is obtained by comparison with a
             branching random walk killed to the left of a linear
             boundary and a finite branchingselection particle system. ©
             2010 Elsevier B.V. All rights reserved.},
   Doi = {10.1016/j.spa.2010.03.011},
   Key = {fds243524}
}

@article{fds243557,
   Author = {Durrett, R},
   Title = {Some features of the spread of epidemics and information on
             a random graph.},
   Journal = {Proceedings of the National Academy of Sciences of the
             United States of America},
   Volume = {107},
   Number = {10},
   Pages = {4491-4498},
   Year = {2010},
   Month = {March},
   ISSN = {0027-8424},
   url = {http://dx.doi.org/10.1073/pnas.0914402107},
   Abstract = {Random graphs are useful models of social and technological
             networks. To date, most of the research in this area has
             concerned geometric properties of the graphs. Here we focus
             on processes taking place on the network. In particular we
             are interested in how their behavior on networks differs
             from that in homogeneously mixing populations or on regular
             lattices of the type commonly used in ecological
             models.},
   Doi = {10.1073/pnas.0914402107},
   Key = {fds243557}
}

@article{fds243525,
   Author = {Durrett, R and Moseley, S},
   Title = {Evolution of resistance and progression to disease during
             clonal expansion of cancer.},
   Journal = {Theoretical Population Biology},
   Volume = {77},
   Number = {1},
   Pages = {42-48},
   Year = {2010},
   Month = {February},
   ISSN = {0040-5809},
   url = {http://dx.doi.org/10.1016/j.tpb.2009.10.008},
   Abstract = {Inspired by previous work of Iwasa et al. (2006) and Haeno
             et al. (2007), we consider an exponentially growing
             population of cancerous cells that will evolve resistance to
             treatment after one mutation or display a disease phenotype
             after two or more mutations. We prove results about the
             distribution of the first time when k mutations have
             accumulated in some cell, and about the growth of the number
             of type-k cells. We show that our results can be used to
             derive the previous results about a tumor grown to a fixed
             size.},
   Doi = {10.1016/j.tpb.2009.10.008},
   Key = {fds243525}
}

@article{fds243530,
   Author = {Durrett, R and Foo, J and Leder, K and Mayberry, J and Michor,
             F},
   Title = {Evolutionary dynamics of tumor progression with random
             fitness values},
   Journal = {Theoretical Population Biology},
   Volume = {78},
   Number = {1},
   Pages = {54-66},
   Year = {2010},
   ISSN = {0040-5809},
   url = {http://dx.doi.org/10.1016/j.tpb.2010.05.001},
   Abstract = {Most human tumors result from the accumulation of multiple
             genetic and epigenetic alterations in a single cell.
             Mutations that confer a fitness advantage to the cell are
             known as driver mutations and are causally related to
             tumorigenesis. Other mutations, however, do not change the
             phenotype of the cell or even decrease cellular fitness.
             While much experimental effort is being devoted to the
             identification of the functional effects of individual
             mutations, mathematical modeling of tumor progression
             generally considers constant fitness increments as mutations
             are accumulated. In this paper we study a mathematical model
             of tumor progression with random fitness increments. We
             analyze a multi-type branching process in which cells
             accumulate mutations whose fitness effects are chosen from a
             distribution. We determine the effect of the fitness
             distribution on the growth kinetics of the tumor. This work
             contributes to a quantitative understanding of the
             accumulation of mutations leading to cancer. © 2010
             Elsevier Inc.},
   Doi = {10.1016/j.tpb.2010.05.001},
   Key = {fds243530}
}

@article{fds243515,
   Author = {Chatterjee, S and Durrett, R},
   Title = {Contact processes on random graphs with power law degree
             distributions have critical value 0},
   Journal = {The Annals of Probability},
   Volume = {37},
   Number = {6},
   Pages = {2332-2356},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2009},
   Month = {November},
   ISSN = {0091-1798},
   url = {http://dx.doi.org/10.1214/09-AOP471},
   Abstract = {If we consider the contact process with infection rate λ on
             a random graph on n vertices with power law degree
             distributions, mean field calculations suggest that the
             critical value λc of the infection rate is positive if the
             power α>3. Physicists seem to regard this as an established
             fact, since the result has recently been generalized to
             bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl.
             Acad. Sci. USA 105 (2008) 1399-1404]. Here, we show that the
             critical value λc is zero for any value of α>3, and the
             contact process starting from all vertices infected, with a
             probability tending to 1 as n →∞, maintains a positive
             density of infected sites for time at least exp(n1-δ) for
             any δ>0. Using the last result, together with the contact
             process duality, we can establish the existence of a
             quasi-stationary distribution in which a randomly chosen
             vertex is occupied with probability ρ(λ). It is expected
             that ρ(λ)~ Cλβ as λ → 0. Here we show that α - 1 ≤
             β ≤ 2α - 3, and so β>2 for α>3. Thus even though the
             graph is locally tree-like, β does not take the mean field
             critical value β = 1. © Institute of Mathematical
             Statistics, 2009.},
   Doi = {10.1214/09-AOP471},
   Key = {fds243515}
}

@article{fds304477,
   Author = {Chan, B and Durrett, R and Lanchier, N},
   Title = {Coexistence for a multitype contact process with
             seasons},
   Journal = {The Annals of Applied Probability},
   Volume = {19},
   Number = {5},
   Pages = {1921-1943},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2009},
   Month = {October},
   ISSN = {1050-5164},
   url = {http://dx.doi.org/10.1214/09-aap599},
   Abstract = {We introduce a multitype contact process with temporal
             heterogeneity involving two species competing for space on
             the d-dimensional integer lattice. Time is divided into
             seasons called alternately season 1 and season 2. We prove
             that there is an open set of the parameters for which both
             species can coexist when their dispersal range is large
             enough. Numerical simulations also suggest that three
             species can coexist in the presence of two seasons. This
             contrasts with the long-term behavior of the
             time-homogeneous multitype contact process for which the
             species with the higher birth rate outcompetes the other
             species when the death rates are equal. © Institute of
             Mathematical Statistics, 2009.},
   Doi = {10.1214/09-aap599},
   Key = {fds304477}
}

@article{fds243539,
   Author = {Durrett, R and Remenik, D},
   Title = {Chaos in a spatial epidemic model},
   Journal = {The Annals of Applied Probability},
   Volume = {19},
   Number = {4},
   Pages = {1656-1685},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2009},
   Month = {August},
   ISSN = {1050-5164},
   MRCLASS = {60K35 (37D45 37N25 60J10 92D30)},
   MRNUMBER = {2538084 (2010k:60322)},
   url = {http://dx.doi.org/10.1214/08-aap581},
   Abstract = {We investigate an interacting particle system inspired by
             the gypsy moth, whose populations grow until they become
             sufficiently dense so that an epidemic reduces them to a low
             level. We consider this process on a random 3-regular graph
             and on the d-dimensional lattice and torus, with d = 2. On
             the finite graphs with global dispersal or with a dispersal
             radius that grows with the number of sites, we prove
             convergence to a dynamical system that is chaotic for some
             parameter values. We conjecture that on the infinite lattice
             with a fixed finite dispersal distance, distant parts of the
             lattice oscillate out of phase so there is a unique
             nontrivial stationary distribution. © Institute of
             Mathematical Statistics, 2009.},
   Doi = {10.1214/08-aap581},
   Key = {fds243539}
}

@article{fds243555,
   Author = {Durrett, R},
   Title = {Coexistence in stochastic spatial models},
   Journal = {The Annals of Applied Probability},
   Volume = {19},
   Number = {2},
   Pages = {477-496},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2009},
   Month = {April},
   ISSN = {1050-5164},
   MRCLASS = {60K35 (92D25)},
   MRNUMBER = {MR2521876 (2010g:60213)},
   url = {http://dx.doi.org/10.1214/08-aap590},
   Abstract = {In this paper I will review twenty years of work on the
             question: When is there coexistence in stochastic spatial
             models? The answer, announced in Durrett and Levin [Theor.
             Pop. Biol. 46 (1994) 363-394], and that we explain in this
             paper is that this can be determined by examining the
             mean-field ODE. There are a number of rigorous results in
             support of this picture, but we will state nine challenging
             and important open problems, most of which date from the
             1990's. © Institute of Mathematical Statistics,
             2009.},
   Doi = {10.1214/08-aap590},
   Key = {fds243555}
}

@article{fds243556,
   Author = {Durrett, R and Schmidt, D and Schweinsberg, J},
   Title = {A waiting time problem arising from the study of multi-stage
             carcinogenesis},
   Journal = {The Annals of Applied Probability},
   Volume = {19},
   Number = {2},
   Pages = {676-718},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2009},
   Month = {April},
   ISSN = {1050-5164},
   MRCLASS = {60J80 (60J25 60K40 92C50)},
   MRNUMBER = {MR2521885 (2010f:60243)},
   url = {http://dx.doi.org/10.1214/08-aap559},
   Abstract = {We consider the population genetics problem: how long does
             it take before some member of the population has m specified
             mutations? The case m = 2 is relevant to onset of cancer due
             to the inactivation of both copies of a tumor suppressor
             gene. Models for larger m are needed for colon cancer and
             other diseases where a sequence of mutations leads to cells
             with uncontrolled growth. © Institute of Mathematical
             Statistics, 2009.},
   Doi = {10.1214/08-aap559},
   Key = {fds243556}
}

@article{fds243512,
   Author = {Durrett, R and Schmidt, D},
   Title = {Reply to Michael Behe},
   Journal = {Genetics},
   Volume = {181},
   Number = {2},
   Pages = {821-822},
   Publisher = {Genetics Society of America},
   Year = {2009},
   Month = {February},
   ISSN = {0016-6731},
   url = {http://dx.doi.org/10.1534/genetics.109.100800},
   Doi = {10.1534/genetics.109.100800},
   Key = {fds243512}
}

@article{fds243538,
   Author = {Durrett, R and Popovic, L},
   Title = {Degenerate diffusions arising from gene duplication
             models},
   Journal = {The Annals of Applied Probability},
   Volume = {19},
   Number = {1},
   Pages = {15-48},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2009},
   Month = {February},
   ISSN = {1050-5164},
   MRNUMBER = {MR2521876 (2010g:60213)},
   url = {http://dx.doi.org/10.1214/08-aap530},
   Abstract = {We consider two processes that have been used to study gene
             duplication, Watterson's [Genetics 105 (1983) 745-766]
             double recessive null model and Lynch and Force's [Genetics
             154 (2000) 459-473] subfunctionalization model. Though the
             state spaces of these diffusions are two and
             six-dimensional, respectively, we show in each case that the
             diffusion stays close to a curve. Using ideas of
             Katzenberger [Ann. Probab. 19 (1991) 1587-1628] we show that
             one-dimensional projections converge to diffusion processes,
             and we obtain asymptotics for the time to loss of one gene
             copy. As a corollary we find that the probability of
             subfunctionalization decreases exponentially fast as the
             population size increases. This rigorously confirms a result
             Ward and Durrett [Theor. Pop. Biol. 66 (2004) 93-100] found
             by simulation that the likelihood of subfunctionalization
             for gene duplicates decays exponentially fast as the
             population size increases. © Institute of Mathematical
             Statistics, 2009.},
   Doi = {10.1214/08-aap530},
   Key = {fds243538}
}

@article{fds243514,
   Author = {Wu, F and Eannetta, NT and Xu, Y and Durrett, R and Mazourek, M and Jahn,
             MM and Tanksley, SD},
   Title = {A COSII genetic map of the pepper genome provides a detailed
             picture of synteny with tomato and new insights into recent
             chromosome evolution in the genus Capsicum},
   Journal = {Tag Theoretical and Applied Genetics},
   Volume = {118},
   Number = {7},
   Pages = {1279-1293},
   Year = {2009},
   ISSN = {0040-5752},
   url = {http://dx.doi.org/10.1007/s00122-009-0980-y},
   Abstract = {We report herein the development of a pepper genetic linkage
             map which comprises 299 orthologous markers between the
             pepper and tomato genomes (including 263 conserved ortholog
             set II or COSII markers). The expected position of
             additional 288 COSII markers was inferred in the pepper map
             via pepper-tomato synteny, bringing the total orthologous
             markers in the pepper genome to 587. While pepper maps have
             been previously reported, this is the first complete map in
             the sense that all markers could be placed in 12 linkage
             groups corresponding to the 12 chromosomes. The map
             presented herein is relevant to the genomes of cultivated C.
             annuum and wild C. annuum (as well as related Capsicum
             species) which differ by a reciprocal chromosome
             translocation. This map is also unique in that it is largely
             based on COSII markers, which permits the inference of a
             detailed syntenic relationship between the pepper and tomato
             genomes-shedding new light on chromosome evolution in the
             Solanaceae. Since divergence from their last common ancestor
             is approximately 20 million years ago, the two genomes have
             become differentiated by a minimum number of 19 inversions
             and 6 chromosome translocations, as well as numerous
             putative single gene transpositions. Nevertheless, the two
             genomes share 35 conserved syntenic segments (CSSs) within
             which gene/marker order is well preserved. The high
             resolution COSII synteny map described herein provides a
             platform for cross-reference of genetic and genomic
             information (including the tomato genome sequence) between
             pepper and tomato and therefore will facilitate both applied
             and basic research in pepper. © 2009 Springer-Verlag.},
   Doi = {10.1007/s00122-009-0980-y},
   Key = {fds243514}
}

@article{fds243542,
   Author = {Chan, B and Durrett, R and Lanchier, N},
   Title = {Coexistence in a particle system with seasons},
   Journal = {Ann. Appl. Probab.},
   Volume = {19},
   Number = {5},
   Pages = {1921-1943},
   Year = {2009},
   ISSN = {1050-5164},
   url = {http://dx.doi.org/10.1214/09-AAP599},
   Abstract = {We introduce a multitype contact process with temporal
             heterogeneity involving two species competing for space on
             the d-dimensional integer lattice. Time is divided into
             seasons called alternately season 1 and season 2. We prove
             that there is an open set of the parameters for which both
             species can coexist when their dispersal range is large
             enough. Numerical simulations also suggest that three
             species can coexist in the presence of two seasons. This
             contrasts with the long-term behavior of the
             time-homogeneous multitype contact process for which the
             species with the higher birth rate outcompetes the other
             species when the death rates are equal. © Institute of
             Mathematical Statistics, 2009.},
   Doi = {10.1214/09-AAP599},
   Key = {fds243542}
}

@article{fds243554,
   Author = {Durrett, R and Lanchier, N},
   Title = {Coexistence in host–pathogen systems},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {118},
   Number = {6},
   Pages = {1004-1021},
   Publisher = {Elsevier BV},
   Year = {2008},
   Month = {June},
   ISSN = {0304-4149},
   MRCLASS = {60K35 (92D25)},
   MRNUMBER = {MR2418255 (2009g:60131)},
   url = {http://dx.doi.org/10.1016/j.spa.2007.07.008},
   Abstract = {Lanchier and Neuhauser have initiated the study of
             host-symbiont systems but have concentrated on the case in
             which the birth rates for unassociated hosts are equal. Here
             we allow the birth rates to be different and identify cases
             in which a host with a specialist pathogen can coexist with
             a second species. Our calculations suggest that it is
             possible for two hosts with specialist pathogens to coexist
             but it is not possible for a host with a specialist
             mutualist to coexist with a second species.},
   Doi = {10.1016/j.spa.2007.07.008},
   Key = {fds243554}
}

@article{fds243553,
   Author = {Durrett, R and Restrepo, M},
   Title = {One-dimensional stepping stone models, sardine genetics and
             Brownian local time},
   Journal = {The Annals of Applied Probability},
   Volume = {18},
   Number = {1},
   Pages = {334-358},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2008},
   Month = {February},
   ISSN = {1050-5164},
   MRCLASS = {60K35 (60J55 92D10)},
   MRNUMBER = {MR2380901 (2008j:60229)},
   url = {http://dx.doi.org/10.1214/07-aap451},
   Abstract = {Consider a one-dimensional stepping stone model with
             colonies of size M and per-generation migration probability
             v, or a voter model on ℤ in which interactions occur over
             a distance of order K. Sample one individual at the origin
             and one at L. We show that if Mv/L and L/K2 converge to
             positive finite limits, then the genealogy of the sample
             converges to a pair of Brownian motions that coalesce after
             the local time of their difference exceeds an independent
             exponentially distributed random variable. The computation
             of the distribution of the coalescence time leads to a
             one-dimensional parabolic differential equation with an
             interesting boundary condition at 0. © Institute of
             Mathematical Statistics, 2008.},
   Doi = {10.1214/07-aap451},
   Key = {fds243553}
}

@article{fds243540,
   Author = {Berestycki, N and Durrett, R},
   Title = {Limiting behavior for the distance of a random
             walk},
   Journal = {Electronic Journal of Probability},
   Volume = {13},
   Pages = {374-395},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2008},
   Month = {January},
   ISSN = {1083-6489},
   MRCLASS = {60G50 (60C05 60J10)},
   MRNUMBER = {MR2386737 (2009d:60130)},
   url = {http://dx.doi.org/10.1214/EJP.v13-490},
   Abstract = {In this paper we study some aspects of the behavior of
             random walks on large but finite graphs before they have
             reached their equilibrium distribution. This investigation
             is motivated by a result we proved recently for the random
             transposition random walk: the distance from the starting
             point of the walk has a phase transition from a linear
             regime to a sublinear regime at time n/2. Here, we study the
             examples of random 3-regular graphs, random adjacent
             transpositions, and riffle shuffles. In the case of a random
             3-regular graph, there is a phase transition where the speed
             changes from 1/3 to 0 at time 3 log2 n. A similar result is
             proved for riffle shuffles, where the speed changes from 1
             to 0 at time log2 n. Both these changes occur when a
             distance equal to the average diameter of the graph is
             reached. However in the case of random adjacent
             transpositions, the behavior is more complex. We find that
             there is no phase transition, even though the distance has
             different scalings in three different regimes. © 2008
             Applied Probability Trust.},
   Doi = {10.1214/EJP.v13-490},
   Key = {fds243540}
}

@article{fds323655,
   Author = {Chung, KL and Durrett, R},
   Title = {Downcrossings and local time},
   Pages = {585-587},
   Booktitle = {Selected Works of Kai Lai Chung},
   Publisher = {World Scientific},
   Year = {2008},
   Month = {January},
   ISBN = {9789812833853},
   url = {http://dx.doi.org/10.1142/9789812833860_0037},
   Abstract = {© 2008 by World Scientific Publishing Co. Pte. Ltd. All
             Rights Reserved. Let formula presented be the standard
             Brownian motion with all paths continuous. Let formula
             presented be the maximum process and formula presented be
             reflecting Brownian motion. If formula presented is the
             number of є to 0 times Y crosses down from e to 0 before
             time t, then it was Paul Lévy's idea thatformula
             presented},
   Doi = {10.1142/9789812833860_0037},
   Key = {fds323655}
}

@article{fds243536,
   Author = {Huerta-Sanchez, E and Durrett, R and Bustamante,
             CD},
   Title = {Population genetics of polymorphism and divergence under
             fluctuating selection},
   Journal = {Genetics},
   Volume = {178},
   Number = {1},
   Pages = {325-337},
   Year = {2008},
   ISSN = {0016-6731},
   url = {http://dx.doi.org/10.1534/genetics.107.073361},
   Abstract = {Current methods for detecting fluctuating selection require
             time series data on genotype frequencies. Here, we propose
             an alternative approach that makes use of DNA polymorphism
             data from a sample of individuals collected at a single
             point in time. Our method uses classical diffusion
             approximations to model temporal fluctuations in the
             selection coefficients to find the expected distribution of
             mutation frequencies in the population. Using the Poisson
             random-field setting we derive the site-frequency spectrum
             (SFS) for three different models of fluctuating selection.
             We find that the general effect of fluctuating selection is
             to produce a more "U"-shaped site-frequency spectrum with an
             excess of high-frequency derived mutations at the expense of
             middle-frequency variants. We present likelihood-ratio
             tests, comparing the fluctuating selection models to the
             neutral model using SFS data, and use Monte Carlo
             simulations to assess their power. We find that we have
             sufficient power to reject a neutral hypothesis using
             samples on the order of a few hundred SNPs and a sample size
             of ∼20 and power to distinguish between selection that
             varies in time and constant selection for a sample of size
             20. We also find that fluctuating selection increases the
             probability of fixation of selected sites even if, on
             average, there is no difference in selection among a pair of
             alleles segregating at the locus. Fluctuating selection
             will, therefore, lead to an increase in the ratio of
             divergence to polymorphism similar to that observed under
             positive directional selection. Copyright © 2008 by the
             Genetics Society of America.},
   Doi = {10.1534/genetics.107.073361},
   Key = {fds243536}
}

@article{fds243541,
   Author = {Durrett, R and Schmidt, D},
   Title = {Waiting for two mutations: With applications to regulatory
             sequence evolution and the limits of Darwinian
             evolution},
   Journal = {Genetics},
   Volume = {180},
   Number = {3},
   Pages = {1501-1509},
   Year = {2008},
   ISSN = {0016-6731},
   url = {http://dx.doi.org/10.1534/genetics.107.082610},
   Abstract = {Results of Nowak and collaborators concerning the onset of
             cancer due to the inactivation of tumor suppressor genes
             give the distribution of the time until some individual in a
             population has experienced two prespecified mutations and
             the time until this mutant phenotype becomes fixed in the
             population. In this article we apply these results to obtain
             insights into regulatory sequence evolution in Drosophila
             and humans. In particular, we examine the waiting time for a
             pair of mutations, the first of which inactivates an
             existing transcription factor binding site and the second of
             which creates a new one. Consistent with recent experimental
             observations for Drosophila, we find that a few million
             years is sufficient, but for humans with a much smaller
             effective population size, this type of change would take
             &gt;100 million years. In addition, we use these results to
             expose flaws in some of Michael Behe's arguments concerning
             mathematical limits to Darwinian evolution. Copyright ©
             2008 by the Genetics Society of America.},
   Doi = {10.1534/genetics.107.082610},
   Key = {fds243541}
}

@article{fds243551,
   Author = {Durrett, R and Zähle, I},
   Title = {On the width of hybrid zones},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {117},
   Number = {12},
   Pages = {1751-1763},
   Publisher = {Elsevier BV},
   Year = {2007},
   Month = {December},
   ISSN = {0304-4149},
   MRCLASS = {60K35 (60J70 92D25)},
   MRNUMBER = {MR2437727 (2010d:60215)},
   url = {http://dx.doi.org/10.1016/j.spa.2006.05.017},
   Abstract = {Hybrid zones occur when two species are found in close
             proximity and interbreeding occurs, but the species'
             characteristics remain distinct. These systems have been
             treated in the biology literature using partial differential
             equations models. Here we investigate a stochastic spatial
             model and prove the existence of a stationary distribution
             that represents the hybrid zone in equilibrium. We calculate
             the width of the hybrid zone, which agrees with the PDE
             formula only in dimensions d ≥ 3. Our results also give
             insight into properties of hybrid zones in patchy
             environments. © 2007 Elsevier Ltd. All rights
             reserved.},
   Doi = {10.1016/j.spa.2006.05.017},
   Key = {fds243551}
}

@article{fds243552,
   Author = {Durrett, R and Jung, P},
   Title = {Two phase transitions for the contact process on small
             worlds},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {117},
   Number = {12},
   Pages = {1910-1927},
   Publisher = {Elsevier BV},
   Year = {2007},
   Month = {December},
   ISSN = {0304-4149},
   MRCLASS = {60K35 (82C22)},
   MRNUMBER = {MR2437735 (2009h:60162)},
   url = {http://dx.doi.org/10.1016/j.spa.2007.03.003},
   Abstract = {In our version of Watts and Strogatz's small world model,
             space is a d-dimensional torus in which each individual has
             in addition exactly one long-range neighbor chosen at random
             from the grid. This modification is natural if one thinks of
             a town where an individual's interactions at school, at
             work, or in social situations introduce long-range
             connections. However, this change dramatically alters the
             behavior of the contact process, producing two phase
             transitions. We establish this by relating the small world
             to an infinite "big world" graph where the contact process
             behavior is similar to the contact process on a tree. We
             then consider the contact process on a slightly modified
             small world model in order to show that its behavior is
             decidedly different from that of the contact process on a
             tree. © 2007 Elsevier Ltd. All rights reserved.},
   Doi = {10.1016/j.spa.2007.03.003},
   Key = {fds243552}
}

@article{fds243537,
   Author = {York, TL and Durrett, R and Nielsen, R},
   Title = {Dependence of paracentric inversion rate on tract
             length.},
   Journal = {Bmc Bioinformatics},
   Volume = {8},
   Pages = {115},
   Year = {2007},
   Month = {April},
   ISSN = {1471-2105},
   url = {http://dx.doi.org/10.1186/1471-2105-8-115},
   Abstract = {BACKGROUND:We develop a Bayesian method based on MCMC for
             estimating the relative rates of pericentric and paracentric
             inversions from marker data from two species. The method
             also allows estimation of the distribution of inversion
             tract lengths. RESULTS:We apply the method to data from
             Drosophila melanogaster and D. yakuba. We find that
             pericentric inversions occur at a much lower rate compared
             to paracentric inversions. The average paracentric inversion
             tract length is approx. 4.8 Mb with small inversions being
             more frequent than large inversions. If the two breakpoints
             defining a paracentric inversion tract are uniformly and
             independently distributed over chromosome arms there will be
             more short tract-length inversions than long; we find an
             even greater preponderance of short tract lengths than this
             would predict. Thus there appears to be a correlation
             between the positions of breakpoints which favors shorter
             tract lengths. CONCLUSION:The method developed in this paper
             provides the first statistical estimator for estimating the
             distribution of inversion tract lengths from marker data.
             Application of this method for a number of data sets may
             help elucidate the relationship between the length of an
             inversion and the chance that it will get
             accepted.},
   Doi = {10.1186/1471-2105-8-115},
   Key = {fds243537}
}

@article{fds243550,
   Author = {Durrett, R and Schmidt, D},
   Title = {Waiting for regulatory sequences to appear},
   Journal = {The Annals of Applied Probability},
   Volume = {17},
   Number = {1},
   Pages = {1-32},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2007},
   Month = {February},
   ISSN = {1050-5164},
   MRCLASS = {92D10 (60C05 60F05 92D20)},
   MRNUMBER = {MR2292578 (2007j:92034)},
   url = {http://dx.doi.org/10.1214/105051606000000619},
   Abstract = {One possible explanation for the substantial organismal
             differences between humans and chimpanzees is that there
             have been changes in gene regulation. Given what is known
             about transcription factor binding sites, this motivates the
             following probability question: given a 1000 nucleotide
             region in our genome, how long does it take for a specified
             six to nine letter word to appear in that region in some
             individual? Stone and Wray [Mol. Biol. Evol. 18 (2001)
             1764-1770] computed 5,950 years as the answer for six letter
             words. Here, we will show that for words of length 6, the
             average waiting time is 100,000 years, while for words of
             length 8, the waiting time has mean 375,000 years when there
             is a 7 out of 8 letter match in the population consensus
             sequence (an event of probability roughly 5/16) and has mean
             650 million years when there is not. Fortunately, in
             biological reality, the match to the target word does not
             have to be perfect for binding to occur. If we model this by
             saying that a 7 out of 8 letter match is good enough, the
             mean reduces to about 60,000 years. © Institute of
             Mathematical Statistics, 2007.},
   Doi = {10.1214/105051606000000619},
   Key = {fds243550}
}

@article{fds183998,
   Author = {R. Durrett and Iljana Zahle},
   Title = {On the width of hybrid zones},
   Journal = {Stochastic Processes and their Applications},
   Volume = {117},
   Pages = {1751--1763},
   Year = {2007},
   ISSN = {0304-4149},
   MRCLASS = {60K35 (60J70 92D25)},
   MRNUMBER = {2437727 (2010d:60215)},
   url = {http://www.ams.org/mathscinet-getitem?mr=2010d:60215},
   Key = {fds183998}
}

@article{fds243531,
   Author = {Huerta-Sanchez, E and Durrett, R},
   Title = {Wagner's canalization model},
   Journal = {Theoretical Population Biology},
   Volume = {71},
   Number = {2},
   Pages = {121-130},
   Year = {2007},
   ISSN = {0040-5809},
   url = {http://dx.doi.org/10.1016/j.tpb.2006.10.006},
   Abstract = {Wagner (1996, Does evolutionary plasticity evolve? Evolution
             50, 1008-1023.) and Siegal and Bergman (2002, Waddington's
             canalization revisited: Developmental stability and
             evolution. Proc. Natl. Acad. Sci. USA 99, 10528-10532.) have
             studied a simple model of the evolution of a network of N
             genes, in order to explain the observed phenomenon that
             systems evolve to be robust. These authors primarily
             considered the case N = 10 and used simulations to reach
             their conclusions. Here we investigate this model in more
             detail, considering systems of different sizes with and
             without recombination, and with selection for convergence
             instead of to a specified limit. For the simpler
             evolutionary model lacking recombination, we analyze the
             system as a neutral network. This allows us to describe the
             equilibrium distribution networks within genotype space. Our
             results show that, given a sufficiently large population
             size, the qualitative observation that systems evolve to be
             robust, is itself robust, as it does not depend on the
             details of the model. In simple terms, robust systems have
             more viable offspring, so the evolution of robustness is
             merely selection for increased fecundity, an observation
             that is well known in the theory of neutral networks. ©
             2006 Elsevier Inc. All rights reserved.},
   Doi = {10.1016/j.tpb.2006.10.006},
   Key = {fds243531}
}

@article{fds243533,
   Author = {De, A and Durrett, R},
   Title = {Spatial structure of the human population contributes to the
             slow decay of linkage diseqeuilibrium and shifts the site
             frequency spectrum},
   Journal = {Genetics},
   Volume = {176},
   Number = {2},
   Pages = {969-981},
   Year = {2007},
   ISSN = {0016-6731},
   url = {http://dx.doi.org/10.1534/genetics.107.071464},
   Abstract = {The symmetric island model with D demes and equal migration
             rates is often chosen for the investigation of the
             consequences of population subdivision. Here we show that a
             stepping-stone model has a more pronounced effect on the
             genealogy of a sample. For samples from a small geographical
             region commonly used in genetic studies of humans and
             Drosophila, there is a shift of the frequency spectrum that
             decreases the number of low-frequency-derived alleles and
             skews the distribution of statistics of Tajima, Fu and Li,
             and Fay and Wu. Stepping-stone spatial structure also
             changes the two-locus sampling distribution and increases
             both linkage disequilibrium and the probability that two
             sites are perfectly correlated. This may cause a false
             prediction of cold spots of recombination and may confuse
             haplotype tests that compute probabilities on the basis of a
             homogeneously mixing population. Copyright © 2007 by the
             Genetics Society of America.},
   Doi = {10.1534/genetics.107.071464},
   Key = {fds243533}
}

@article{fds243535,
   Author = {Sainudiin, R and Clark, AG and Durrett, RT},
   Title = {Simple models of genomic variation in human SNP
             density},
   Journal = {Bmc Genomics},
   Volume = {8},
   Year = {2007},
   ISSN = {1471-2164},
   url = {http://dx.doi.org/10.1186/1471-2164-8-146},
   Abstract = {Background: Descriptive hierarchical Poisson models and
             population-genetic coalescent mixture models are used to
             describe the observed variation in single-nucleotide
             polymorphism (SNP) density from samples of size two across
             the human genome. Results: Using empirical estimates of
             recombination rate across the human genome and the observed
             SNP density distribution, we produce a maximum likelihood
             estimate of the genomic heterogeneity in the scaled mutation
             rate θ. Such models produce significantly better fits to
             the observed SNP density distribution than those that ignore
             the empirically observed recombinational heterogeneities.
             Conclusion: Accounting for mutational and recombinational
             heterogeneities can allow for empirically sound null
             distributions in genome scans for "outliers", when the
             alternative hypotheses include fundamentally historical and
             unobserved phenomena. © 2007 Sainudiin et al; licensee
             BioMed Central Ltd.},
   Doi = {10.1186/1471-2164-8-146},
   Key = {fds243535}
}

@article{fds304476,
   Author = {De, A and Durrett, R},
   Title = {Stepping-stone spatial structure causes slow decay of
             linkage disequilibrium and shifts the site frequency
             spectrum},
   Journal = {Genetics},
   Volume = {176},
   Number = {2},
   Pages = {969-981},
   Year = {2007},
   ISSN = {0016-6731},
   url = {http://dx.doi.org/10.1534/genetics.107.071464},
   Abstract = {The symmetric island model with D demes and equal migration
             rates is often chosen for the investigation of the
             consequences of population subdivision. Here we show that a
             stepping-stone model has a more pronounced effect on the
             genealogy of a sample. For samples from a small geographical
             region commonly used in genetic studies of humans and
             Drosophila, there is a shift of the frequency spectrum that
             decreases the number of low-frequency-derived alleles and
             skews the distribution of statistics of Tajima, Fu and Li,
             and Fay and Wu. Stepping-stone spatial structure also
             changes the two-locus sampling distribution and increases
             both linkage disequilibrium and the probability that two
             sites are perfectly correlated. This may cause a false
             prediction of cold spots of recombination and may confuse
             haplotype tests that compute probabilities on the basis of a
             homogeneously mixing population. Copyright © 2007 by the
             Genetics Society of America.},
   Doi = {10.1534/genetics.107.071464},
   Key = {fds304476}
}

@article{fds243548,
   Author = {Berestycki, N and Durrett, R},
   Title = {A phase transition in the random transposition random
             walk},
   Journal = {Probability Theory and Related Fields},
   Volume = {136},
   Number = {2},
   Pages = {203-233},
   Publisher = {Springer Nature},
   Year = {2006},
   Month = {October},
   ISSN = {0178-8051},
   MRCLASS = {60C05 (60G50)},
   MRNUMBER = {MR2240787 (2007i:60009)},
   url = {http://dx.doi.org/10.1007/s00440-005-0479-7},
   Abstract = {Our work is motivated by Bourque and Pevzner's (2002)
             simulation study of the effectiveness of the parsimony
             method in studying genome rearrangement, and leads to a
             surprising result about the random transposition walk on the
             group of permutations on n elements. Consider this walk in
             continuous time starting at the identity and let D t be the
             minimum number of transpositions needed to go back to the
             identity from the location at time t. D t undergoes a phase
             transition: the distance D cn/2̃ u(c)n, where u is an
             explicit function satisfying u(c)=c/2 for c ≤ 1 and
             u(c)&lt;c/2 for c&gt;1. In addition, we describe the
             fluctuations of D cn/2 about its mean in each of the three
             regimes (subcritical, critical and supercritical). The
             techniques used involve viewing the cycles in the random
             permutation as a coagulation-fragmentation process and
             relating the behavior to the Erdos-Renyi random graph
             model.},
   Doi = {10.1007/s00440-005-0479-7},
   Key = {fds243548}
}

@article{fds243549,
   Author = {Chan, B and Durrett, R},
   Title = {A new coexistence result for competing contact
             processes},
   Journal = {The Annals of Applied Probability},
   Volume = {16},
   Number = {3},
   Pages = {1155-1165},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2006},
   Month = {August},
   ISSN = {1050-5164},
   MRCLASS = {60K35},
   MRNUMBER = {MR2260060 (2008h:60400)},
   url = {http://dx.doi.org/10.1214/105051606000000132},
   Abstract = {Neuhauser [Probab. Theory Related Fields 91 (1992) 467-506]
             considered the two-type contact process and showed that on
             ℤ 2 coexistence is not possible if the death rates are
             equal and the particles use the same dispersal neighborhood.
             Here, we show that it is possible for a species with a
             long-,but finite, range dispersal kernel to coexist with a
             superior competitor with nearest-neighbor dispersal in a
             model that includes deaths of blocks due to "forest fires."
             © Institute of Mathematical Statistics,
             2006.},
   Doi = {10.1214/105051606000000132},
   Key = {fds243549}
}

@article{fds243412,
   Author = {Durrett, R},
   Title = {Random graph dynamics},
   Journal = {Random Graph Dynamics},
   Series = {Cambridge Series in Statistical and Probabilistic
             Mathematics},
   Pages = {1-212},
   Publisher = {Cambridge University Press},
   Address = {Cambridge},
   Year = {2006},
   Month = {January},
   ISBN = {978-0-521-86656-9; 0-521-86656-1},
   MRCLASS = {05C80 (05-02 60-02 60C05 60G50 60K35 82C41)},
   MRNUMBER = {MR2271734 (2008c:05167)},
   url = {http://dx.doi.org/10.1017/CBO9780511546594},
   Abstract = {© Rick Durrett 2007 and Cambridge University Press, 2009.
             The theory of random graphs began in the late 1950s in
             several papers by Erdos and Renyi. In the late twentieth
             century, the notion of six degrees of separation, meaning
             that any two people on the planet can be connected by a
             short chain of people who know each other, inspired Strogatz
             and Watts to define the small world random graph in which
             each site is connected to k close neighbors, but also has
             long-range connections. At about the same time, it was
             observed in human social and sexual networks and on the
             Internet that the number of neighbors of an individual or
             computer has a power law distribution. This inspired
             Barabasi and Albert to define the preferential attachment
             model, which has these properties. These two papers have led
             to an explosion of research. While this literature is
             extensive, many of the papers are based on simulations and
             nonrigorous arguments. The purpose of this book is to use a
             wide variety of mathematical argument to obtain insights
             into the properties of these graphs. A unique feature of
             this book is the interest in the dynamics of process taking
             place on the graph in addition to their geometric
             properties, such as connectedness and diameter.},
   Doi = {10.1017/CBO9780511546594},
   Key = {fds243412}
}

@article{fds243534,
   Author = {Durrett, R and Schweinsberg, J},
   Title = {Power laws for family sizes in a duplication
             model},
   Journal = {The Annals of Probability},
   Volume = {33},
   Number = {6},
   Pages = {2094-2126},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2005},
   Month = {November},
   ISSN = {0091-1798},
   MRCLASS = {60J85 (60J80 92D15 92D20)},
   MRNUMBER = {2184092 (2006j:60092)},
   url = {http://dx.doi.org/10.1214/009117905000000369},
   Abstract = {Qian, Luscombe and Gerstein [J. Molecular Biol. 313 (2001)
             673-681] introduced a model of the diversification of
             protein folds in a genome that we may formulate as follows.
             Consider a multitype Yule process starting with one
             individual in which there are no deaths and each individual
             gives birth to a new individual at rate 1. When a new
             individual is born, it has the same type as its parent with
             probability 1 - r and is a new type, different from all
             previously observed types, with probability r. We refer to
             individuals with the same type as families and provide an
             approximation to the joint distribution of family sizes when
             the population size reaches N. We also show that if 1 ≪ S
             ≪ N 1-r, then the number of families of size at least 5 is
             approximately CNS -1/(1-r), while if N 1-r ≪ S the
             distribution decays more rapidly than any power. ©
             Institute of Mathematical Statistics, 2005.},
   Doi = {10.1214/009117905000000369},
   Key = {fds243534}
}

@article{fds243547,
   Author = {Durrett, R and Schweinsberg, J},
   Title = {A coalescent model for the effect of advantageous mutations
             on the genealogy of a population},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {115},
   Number = {10},
   Pages = {1628-1657},
   Publisher = {Elsevier BV},
   Year = {2005},
   Month = {October},
   ISSN = {0304-4149},
   MRCLASS = {92D15 (60J27 92D10 92D20)},
   MRNUMBER = {MR2165337 (2006h:92026)},
   url = {http://dx.doi.org/10.1016/j.spa.2005.04.009},
   Abstract = {When an advantageous mutation occurs in a population, the
             favorable allele may spread to the entire population in a
             short time, an event known as a selective sweep. As a
             result, when we sample n individuals from a population and
             trace their ancestral lines backwards in time, many lineages
             may coalesce almost instantaneously at the time of a
             selective sweep. We show that as the population size goes to
             infinity, this process converges to a coalescent process
             called a coalescent with multiple collisions. A better
             approximation for finite populations can be obtained using a
             coalescent with simultaneous multiple collisions. We also
             show how these coalescent approximations can be used to get
             insight into how beneficial mutations affect the behavior of
             statistics that have been used to detect departures from the
             usual Kingman's coalescent. © 2005 Elsevier B.V. All rights
             reserved.},
   Doi = {10.1016/j.spa.2005.04.009},
   Key = {fds243547}
}

@article{fds243544,
   Author = {Blasiak, J and Durrett, R},
   Title = {Random Oxford graphs},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {115},
   Number = {8},
   Pages = {1257-1278},
   Publisher = {Elsevier BV},
   Year = {2005},
   Month = {August},
   ISSN = {0304-4149},
   MRCLASS = {60C05 (05C80)},
   MRNUMBER = {MR2152374 (2006j:60008)},
   url = {http://dx.doi.org/10.1016/j.spa.2005.03.008},
   Abstract = {Inspired by a concept in comparative genomics, we
             investigate properties of randomly chosen members of G1(m,
             n, t), the set of bipartite graphs with m left vertices, n
             right vertices, t edges, and each vertex of degree at least
             one. We give asymptotic results for the number of such
             graphs and the number of (i, j) trees they contain. We
             compute the thresholds for the emergence of a giant
             component and for the graph to be connected. © 2005
             Elsevier B.V. All rights reserved.},
   Doi = {10.1016/j.spa.2005.03.008},
   Key = {fds243544}
}

@article{fds243545,
   Author = {Schweinsberg, J and Durrett, R},
   Title = {Random partitions approximating the coalescence of lineages
             during a selective sweep},
   Journal = {The Annals of Applied Probability},
   Volume = {15},
   Number = {3},
   Pages = {1591-1651},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2005},
   Month = {August},
   ISSN = {1050-5164},
   MRCLASS = {92D10 (05A18 60C05 60J80 60J85 92D15)},
   MRNUMBER = {MR2152239 (2006c:92012)},
   url = {http://dx.doi.org/10.1214/105051605000000430},
   Abstract = {When a beneficial mutation occurs in a population, the new,
             favored allele may spread to the entire population. This
             process is known as a selective sweep. Suppose we sample n
             individuals at the end of a selective sweep. If we focus on
             a site on the chromosome that is close to the location of
             the beneficial mutation, then many of the lineages will
             likely be descended from the individual that had the
             beneficial mutation, while others will be descended from a
             different individual because of recombination between the
             two sites. We introduce two approximations for the effect of
             a selective sweep. The first one is simple but not very
             accurate: flip n independent coins with probability p of
             heads and say that the lineages whose coins come up heads
             are those that are descended from the individual with the
             beneficial mutation. A second approximation, which is
             related to Kingman's paintbox construction, replaces the
             coin flips by integer-valued random variables and leads to
             very accurate results. © Institute of Mathematical
             Statistics. 2005.},
   Doi = {10.1214/105051605000000430},
   Key = {fds243545}
}

@article{fds243510,
   Author = {Durrett, R and Levin, SA},
   Title = {Can stable social groups be maintained by homophilous
             imitation alone?},
   Journal = {Journal of Economic Behavior and Organization},
   Volume = {57},
   Number = {3},
   Pages = {267-286},
   Publisher = {Elsevier BV},
   Year = {2005},
   Month = {July},
   url = {http://dx.doi.org/10.1016/j.jebo.2003.09.017},
   Abstract = {A central problem in the biological and social sciences
             concerns the conditions required for emergence and
             maintenance of cooperation among unrelated individuals. Most
             models and experiments have been pursued in a game-theoretic
             context and involve reward or punishment. Here, we show that
             such payoffs are unnecessary, and that stable social groups
             can sometimes be maintained provided simply that agents are
             more likely to imitate others who are like them (homophily).
             In contrast to other studies, to sustain multiple types we
             need not impose the restriction that agents also choose to
             make their opinions different from those in other groups. ©
             2004 Elsevier B.V. All rights reserved.},
   Doi = {10.1016/j.jebo.2003.09.017},
   Key = {fds243510}
}

@article{fds243543,
   Author = {Z�hle, I and Cox, JT and Durrett, R},
   Title = {The stepping stone model. II: Genealogies and the infinite
             sites model},
   Journal = {The Annals of Applied Probability},
   Volume = {15},
   Number = {1B},
   Pages = {671-699},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2005},
   Month = {February},
   ISSN = {1050-5164},
   MRCLASS = {60K35 (92D10)},
   MRNUMBER = {MR2114986 (2006d:60157)},
   url = {http://dx.doi.org/10.1214/105051604000000701},
   Abstract = {This paper extends earlier work by Cox and Durrett, who
             studied the coalescence times for two lineages in the
             stepping stone model on the two-dimensional torus. We show
             that the genealogy of a sample of size n is given by a time
             change of Kingman's coalescent. With DNA sequence data in
             mind, we investigate mutation patterns under the infinite
             sites model, which assumes that each mutation occurs at a
             new site. Our results suggest that the spatial structure of
             the human population contributes to the haplotype structure
             and a slower than expected decay of genetic correlation with
             distance revealed by recent studies of the human genome. ©
             Institute of Mathematical Statistics, 2005.},
   Doi = {10.1214/105051604000000701},
   Key = {fds243543}
}

@article{fds177893,
   Author = {R. Durrett},
   Title = {Genome rearrangement},
   Series = {Stat. Biol. Health},
   Pages = {307--323},
   Booktitle = {Statistical methods in molecular evolution},
   Publisher = {Springer},
   Address = {New York},
   Year = {2005},
   MRCLASS = {92D10 (05A05 60K25 60K30 62F15 62P10)},
   MRNUMBER = {MR2161835 (2006f:92021)},
   url = {http://www.ams.org/mathscinet-getitem?mr=2161835},
   Key = {fds177893}
}

@article{fds243532,
   Author = {York, TL and Durrett, RT and Tanksley, S and Nielsen,
             R},
   Title = {Bayesian and maximum likelihood estimation of genetic
             maps},
   Journal = {Genetical Research},
   Volume = {85},
   Number = {2},
   Pages = {159-168},
   Year = {2005},
   url = {http://dx.doi.org/10.1017/S0016672305007494},
   Abstract = {There has recently been increased interest in the use of
             Markov Chain Monte Carlo (MCMC)-based Bayesian methods for
             estimating genetic maps. The advantage of these methods is
             that they can deal accurately with missing data and
             genotyping errors. Here we present an extension of the
             previous methods that makes the Bayesian method applicable
             to large data sets. We present an extensive simulation study
             examining the statistical properties of the method and
             comparing it with the likelihood method implemented in
             Mapmaker. We show that the Maximum A Posteriori (MAP)
             estimator of the genetic distances, corresponding to the
             maximum likelihood estimator, performs better than
             estimators based on the posterior expectation. We also show
             that while the performance is similar between Mapmaker and
             the MCMC-based method in the absence of genotyping errors,
             the MCMC-based method has a distinct advantage in the
             presence of genotyping errors. A similar advantage of the
             Bayesian method was not observed for missing data. We also
             re-analyse a recently published set of data from the
             eggplant and show that the use of the MCMC-based method
             leads to smaller estimates of genetic distances. © 2005
             Cambridge University Press.},
   Doi = {10.1017/S0016672305007494},
   Key = {fds243532}
}

@article{fds243546,
   Author = {Durrett, R and Mytnik, L and Perkins, E},
   Title = {Competing super-Brownian motions as limits of interacting
             particle systems},
   Journal = {Electronic Journal of Probability},
   Volume = {10},
   Pages = {1147-1220},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2005},
   ISSN = {1083-6489},
   MRCLASS = {60G57 (60G17)},
   MRNUMBER = {MR2164042 (2006f:60052)},
   url = {http://dx.doi.org/10.1214/EJP.v10-229},
   Abstract = {We study two-type branching random walks in which the birth
             or death rate of each type can depend on the number of
             neighbors of the opposite type. This competing species model
             contains variants of Durrett's predator-prey model and
             Durrett and Levin's colicin model as special cases. We
             verify in some cases convergence of scaling limits of these
             models to a pair of super-Brownian motions interacting
             through their collision local times, constructed by Evans
             and Perkins.},
   Doi = {10.1214/EJP.v10-229},
   Key = {fds243546}
}

@article{fds243505,
   Author = {Durrett, R and Schweinsberg, J},
   Title = {Approximating selective sweeps},
   Journal = {Theoretical Population Biology},
   Volume = {66},
   Number = {2},
   Pages = {129-138},
   Year = {2004},
   url = {http://dx.doi.org/10.1016/j.tpb.2004.04.002},
   Abstract = {The fixation of advantageous mutations in a population has
             the effect of reducing variation in the DNA sequence near
             that mutation. Kaplan et al. (1989) used a three-phase
             simulation model to study the effect of selective sweeps on
             genealogies. However, most subsequent work has simplified
             their approach by assuming that the number of individuals
             with the advantageous allele follows the logistic
             differential equation. We show that the impact of a
             selective sweep can be accurately approximated by a random
             partition created by a stick-breaking process. Our
             simulation results show that ignoring the randomness when
             the number of individuals with the advantageous allele is
             small can lead to substantial errors. © 2004 Elsevier Inc.
             All rights reserved.},
   Doi = {10.1016/j.tpb.2004.04.002},
   Key = {fds243505}
}

@article{fds243506,
   Author = {Ward, R and Durrett, R},
   Title = {Subfunctionalization: How often does it occur? How long does
             it take?},
   Journal = {Theoretical Population Biology},
   Volume = {66},
   Number = {2},
   Pages = {93-100},
   Year = {2004},
   url = {http://dx.doi.org/10.1016/j.tpb.2004.03.004},
   Abstract = {The mechanisms responsible for the preservation of duplicate
             genes have been debated for more than 70 years. Recently,
             Lynch and Force have proposed a new explanation:
             subfunctionalization - after duplication the two gene copies
             specialize to perform complementary functions. We
             investigate the probability that subfunctionalization
             occurs, the amount of time after duplication that it takes
             for the outcome to be resolved, and the relationship of
             these quantities to the population size and mutation rates.
             © 2004 Elsevier Inc. All rights reserved.},
   Doi = {10.1016/j.tpb.2004.03.004},
   Key = {fds243506}
}

@article{fds243507,
   Author = {Durrett, R and Nielsen, R and York, TL},
   Title = {Bayesian Estimation of Genomic Distance},
   Journal = {Genetics},
   Volume = {166},
   Number = {1},
   Pages = {621-629},
   Year = {2004},
   url = {http://dx.doi.org/10.1534/genetics.166.1.621},
   Abstract = {We present a Bayesian approach to the problem of inferring
             the number of inversions and translocations separating two
             species. The main reason for developing this method is that
             it will allow us to test hypotheses about the underlying
             mechanisms, such as the distribution of inversion track
             lengths or rate constancy among lineages. Here, we apply
             these methods to comparative maps of eggplant and tomato,
             human and cat, and human and cattle with 170, 269, and 422
             markers, respectively. In the first case the most likely
             number of events is larger than the parsimony value. In the
             last two cases the parsimony solutions have very small
             probability.},
   Doi = {10.1534/genetics.166.1.621},
   Key = {fds243507}
}

@article{fds243508,
   Author = {Schmidt, D and Durrett, R},
   Title = {Adaptive evolution drives the diversification of zinc-finger
             binding domains},
   Journal = {Molecular Biology and Evolution},
   Volume = {21},
   Number = {12},
   Pages = {2326-2339},
   Year = {2004},
   url = {http://dx.doi.org/10.1093/molbev/msh246},
   Abstract = {The human genome is estimated to contain 700 zinc-finger
             genes, which perform many key functions, including
             regulating transcription. The dramatic increase in the
             number of these genes as we move from yeast to C. elegans to
             Drosophila and to humans, as well as the clustered
             organization of these genes in humans, suggests that gene
             duplication has played an important role in expanding this
             family of genes. Using likelihood methods developed by Yang
             and parsimony methods introduced by Suzuki and Gojobori, we
             have investigated four clusters of zinc-finger genes on
             human chromosome 19 and found evidence that positive
             selection was involved in diversifying the family of
             zinc-finger binding motifs.},
   Doi = {10.1093/molbev/msh246},
   Key = {fds243508}
}

@article{fds243509,
   Author = {Sainudiin, R and Durrett, RT and Aquadro, CF and Nielsen,
             R},
   Title = {Microsatellite mutation models: Insights from a comparison
             of humans and chimpanzees},
   Journal = {Genetics},
   Volume = {168},
   Number = {1},
   Pages = {383-395},
   Year = {2004},
   url = {http://dx.doi.org/10.1534/genetics.103.022665},
   Abstract = {Using genomic data from homologous microsatellite loci of
             pure AC repeats in humans and chimpanzees, several models of
             microsatellite evolution are tested and compared using
             likelihood-ratio tests and the Akaike information criterion.
             A proportional-rate, linear-biased, one-phase model emerges
             as the best model. A focal length toward which the
             mutational and/or substitutional process is linearly biased
             is a crucial feature of microsatellite evolution. We find
             that two-phase models do not lead to a significantly better
             fit than their one-phase counterparts. The performance of
             models based on the fit of their stationary distributions to
             the empirical distribution of microsatellite lengths in the
             human genome is consistent with that based on the
             human-chimp comparison. Microsatellites interrupted by even
             a single point mutation exhibit a twofold decrease in their
             mutation rate when compared to pure AC repeats. In general,
             models that allow chimps to have a larger per-repeat unit
             slippage rate and/or a shorter focal length compared to
             humans give a better fit to the human-chimp data as well as
             the human genomic data.},
   Doi = {10.1534/genetics.103.022665},
   Key = {fds243509}
}

@article{fds243503,
   Author = {Durrett, R and Limic, V},
   Title = {Rigorous results for the N K model},
   Journal = {The Annals of Probability},
   Volume = {31},
   Number = {4},
   Pages = {1713-1753},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2003},
   Month = {October},
   ISSN = {0091-1798},
   url = {http://dx.doi.org/10.1214/aop/1068646364},
   Abstract = {Motivated by the problem of the evolution of DNA sequences,
             Kauffman and Levin introduced a model in which fitnesses
             were assigned to strings of 0's and 1's of length N based on
             the values observed in a sliding window of length K + 1.
             When K ≥ 1, the landscape is quite complicated with many
             local maxima. Its properties have been extensively
             investigated by simulation but until our work and the
             independent investigations of Evans and Steinsaltz little
             was known rigorously about its properties except in the case
             K = N - 1. Here, we prove results about the number of local
             maxima, their heights and the height of the global maximum.
             Our main tool is the theory of (substochastic) Harris
             chains.},
   Doi = {10.1214/aop/1068646364},
   Key = {fds243503}
}

@article{fds243501,
   Author = {Calabrese, P and Durrett, R},
   Title = {Dinucleotide repeats in the drosophila and human genomes
             have complex, length-dependent mutation processes},
   Journal = {Molecular Biology and Evolution},
   Volume = {20},
   Number = {5},
   Pages = {715-725},
   Year = {2003},
   url = {http://dx.doi.org/10.1093/molbev/msg084},
   Abstract = {We use methods of maximum likelihood estimation to fit
             several microsatellite mutation models to the observed
             length distribution of dinucletoide repeats in the
             Drosophila and human genomes. All simple models are rejected
             by this procedure. Two new models, one with quadratic and
             another with piecewise linear slippage rates, have the best
             fits and agree with recent experimental studies by
             predicting that long microsatellites have a bias toward
             contractions.},
   Doi = {10.1093/molbev/msg084},
   Key = {fds243501}
}

@article{fds243502,
   Author = {Cox, T and Durrett, R},
   Title = {Erratum: The stepping stone model: New formulas expose old
             myths (The Annals of Applied Probability (2002) 12
             (1348-1377))},
   Journal = {The Annals of Applied Probability},
   Volume = {13},
   Number = {2},
   Pages = {816-},
   Year = {2003},
   ISSN = {1050-5164},
   Key = {fds243502}
}

@article{fds243504,
   Author = {Durrett, R},
   Title = {Shuffling Chromosomes},
   Journal = {Journal of Theoretical Probability},
   Volume = {16},
   Number = {3},
   Pages = {725-750},
   Year = {2003},
   ISSN = {0894-9840},
   url = {http://dx.doi.org/10.1023/A:1025676617383},
   Abstract = {The gene order of chromosomes can be rearranged by
             chromosomal inversions that reverse the order of segments.
             Motivated by a comparative study of two Drosophila species,
             we investigate the number of reversals that are needed to
             scramble the gene order when all reversals are equally
             likely and when the segments reversed are never more than L
             genes. In studying this question we prove some new results
             about the convergence to equilibrium of shuffling by
             transposition and the one dimensional simple exclusion
             process.},
   Doi = {10.1023/A:1025676617383},
   Key = {fds243504}
}

@article{fds243498,
   Author = {Durrett, R and Limic, V},
   Title = {A surprising Poisson process arising from a species
             competition model},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {102},
   Number = {2},
   Pages = {301-309},
   Publisher = {Elsevier BV},
   Year = {2002},
   Month = {December},
   ISSN = {0304-4149},
   url = {http://dx.doi.org/10.1016/s0304-4149(02)00209-0},
   Abstract = {Motivated by the work of Tilman (Ecology 75 (1994) 2) and
             May and Nowak (J. Theoret. Biol. 170 (1994) 95) we consider
             a process in which points are inserted randomly into the
             unit interval and a new point kills each point to its left
             independently and with probability a. Intuitively this
             dynamic will create a negative dependence between the number
             of points in adjacent intervals. However, we show that the
             ensemble of points converges to a Poisson process with
             intensity 1/(a(1 - x)), and the number of points at time t
             grows like (log t)/a. © 2002 Elsevier Science B.V. All
             rights reserved.},
   Doi = {10.1016/s0304-4149(02)00209-0},
   Key = {fds243498}
}

@article{fds243500,
   Author = {Cox, JT and Durrett, R},
   Title = {The stepping stone model: New formulas expose old
             myths},
   Journal = {The Annals of Applied Probability},
   Volume = {12},
   Number = {4},
   Pages = {1348-1377},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2002},
   Month = {November},
   url = {http://dx.doi.org/10.1214/aoap/1037125866},
   Abstract = {We study the stepping stone model on the two-dimensional
             torus. We prove several new hitting time results for random
             walks from which we derive some simple approximation
             formulas for the homozygosity in the stepping stone model as
             a function of the separation of the colonies and for
             Wright's genetic distance FST. These results confirm a
             result of Crow and Aoki (1984) found by simulation: in the
             usual biological range of parameters FST grows like the log
             of the number of colonies. In the other direction, our
             formulas show that there is significant spatial structure in
             parts of parameter space where Maruyama and Nei (1971) and
             Slatkin and Barton (1989) have called the stepping model
             "effectively panmictic".},
   Doi = {10.1214/aoap/1037125866},
   Key = {fds243500}
}

@article{fds243497,
   Author = {Durrett, R and Kesten, H and Limic, V},
   Title = {Once edge-reinforced random walk on a tree},
   Journal = {Probability Theory and Related Fields},
   Volume = {122},
   Number = {4},
   Pages = {567-592},
   Publisher = {Springer Nature},
   Year = {2002},
   Month = {April},
   url = {http://dx.doi.org/10.1007/s004400100179},
   Abstract = {We consider a nearest neighbor walk on a regular tree, with
             transition probabilities proportional to weights or
             conductances of the edges. Initially all edges have weight
             1, and the weight of an edge is increased to c &gt; 1 when
             the edge is traversed for the first time. After such a
             change the weight of an edge stays at c forever. We show
             that such a walk is transient for all values of c ≥ 1, and
             that the walk moves off to infinity at a linear rate. We
             also prove an invariance principle for the height of the
             walk.},
   Doi = {10.1007/s004400100179},
   Key = {fds243497}
}

@article{fds243496,
   Author = {Balding, DJ and Carothers, AD and Marchini, JL and Cardon, LR and Vetta,
             A and Griffiths, B and Weir, BS and Hill, WG and Goldstein, D and Strimmer,
             K and Myers, S and Beaumont, MA and Glasbey, CA and Mayer, CD and Richardson, S and Marshall, C and Durrett, R and Nielsen, R and Visscher, PM and Knott, SA and Haley, CS and Ball, RD and Hackett, CA and Holmes, S and Husmeier, D and Jansen, RC and Ter Braak and CJF and Maliepaard, CA and Boer, MP and Joyce, P and Li, N and Stephens, M and Marcoulides, GA and Drezner, Z and Mardia, K and McVean, G and Meng, XL and Ochs, MF and Pagel, M and Sha, N and Vannucci, M and Sillanpää, MJ and Sisson, S and Yandell, BS and Jin, C and Satagopan, JM and Gaffney, PJ and Zeng, ZB and Broman, KW and Speed, TP and Fearnhead, P and Donnelly, P and Larget, B and Simon, DL and Kadane, JB and Nicholson, G and Smith, AV and Jónsson, F and Gústafsson, O and Stefánsson, K and Parmigiani, G and Garrett, ES and Anbazhagan, R and Gabrielson, E},
   Title = {Discussion on the meeting on 'statistical modelling and
             analysis of genetic data'},
   Journal = {Journal of the Royal Statistical Society: Series B
             (Statistical Methodology)},
   Volume = {64},
   Number = {4},
   Pages = {737-775},
   Publisher = {WILEY},
   Year = {2002},
   Month = {January},
   ISSN = {1369-7412},
   url = {http://dx.doi.org/10.1111/1467-9868.00359},
   Doi = {10.1111/1467-9868.00359},
   Key = {fds243496}
}

@article{fds243492,
   Author = {Durrett, RT and Chen, K-Y and Tanksley, SD},
   Title = {A simple formula useful for positional cloning},
   Journal = {Genetics},
   Volume = {160},
   Number = {1},
   Pages = {353-355},
   Year = {2002},
   ISSN = {0016-6731},
   Abstract = {We derive a formula for the distribution of the length T of
             the recombination interval containing a target gene and
             using N gametes in a region where R kilobases correspond to
             1 cM. The formula can be used to calculate the number of
             meiotic events required to narrow a target gene down to a
             specific interval size and hence should be useful for
             planning positional cloning experiments. The predictions of
             this formula agree well with the results from a number of
             published experiments in Arabidopsis.},
   Key = {fds243492}
}

@article{fds243494,
   Author = {Durrett, R},
   Title = {Mutual invadability implies coexistence in spatial
             models},
   Journal = {Memoirs of the American Mathematical Society},
   Number = {740},
   Year = {2002},
   Abstract = {In (1994) Durrett and Levin proposed that the equilibrium
             behavior of stochastic spatial models could be determined
             from properties of the solution of the mean field ordinary
             differential equation (ODE) that is obtained by pretending
             that all sites are always independent. Here we prove a
             general result in support of that picture. We give a
             condition on an ordinary differential equation which implies
             that densities stay bounded away from 0 in the associated
             reaction-diffusion equation, and that coexistence occurs in
             the stochastic spatial model with fast stirring. Then using
             biologists' notion of invadability as a guide, we show how
             this condition can be checked in a wide variety of examples
             that involve two or three species: epidemics, diploid
             genetics models, predator-prey systems, and various
             competition models.},
   Key = {fds243494}
}

@article{fds243495,
   Author = {York, TL and Durrett, R and Nielsen, R},
   Title = {Bayesian estimation of the number of inversions in the
             history of two chromosomes},
   Journal = {Journal of Computational Biology},
   Volume = {9},
   Number = {6},
   Pages = {805-818},
   Year = {2002},
   url = {http://dx.doi.org/10.1089/10665270260518281},
   Abstract = {We present a Bayesian approach to the problem of inferring
             the history of inversions separating homologous chromosomes
             from two different species. The method is based on Markov
             Chain Monte Carlo (MCMC) and takes full advantage of all the
             information from marker order. We apply the method both to
             simulated data and to two real data sets. For the simulated
             data, we show that the MCMC method provides accurate
             estimates of the true posterior distributions and in the
             analysis of the real data we show that the most likely
             number of inversions in some cases is considerably larger
             than estimates obtained based on the parsimony inferred
             number of inversions. Indeed, in the case of the Drosophila
             repleta-D. melanogaster comparison, the lower boundary of a
             95% highest posterior density credible interval for the
             number of inversions is considerably larger than the most
             parsimonious number of inversions.},
   Doi = {10.1089/10665270260518281},
   Key = {fds243495}
}

@article{fds243499,
   Author = {Buttel, LA and Durrett, R and Levin, SA},
   Title = {Competition and Species Packing in Patchy
             Environments},
   Journal = {Theoretical Population Biology},
   Volume = {61},
   Number = {3},
   Pages = {265-276},
   Year = {2002},
   url = {http://dx.doi.org/10.1006/tpbi.2001.1569},
   Abstract = {In models of competition in which space is treated as a
             continuum, and population size as continuous, there are no
             limits to the number of species that can coexist. For a
             finite number of sites, N, the results are different. The
             answer will, of course, depend on the model used to ask the
             question. In the Tilman-May-Nowak ordinary differential
             equation model, the number of species is asymptotically C
             log N with most species packed in at the upper end of the
             competitive hierarchy. In contrast, for metapopulation
             models with discrete individuals and stochastic spatial
             systems with various competition neighborhoods, we find a
             traditional species area relationship CNa, with no species
             clumping along the phenotypic gradient. The exponent a is
             larger by a factor of 2 for spatially explicit models. In
             words, a spatial distribution of competitors allows for
             greater diversity than a metapopulation model due to the
             effects of recruitment limitation in their competition. ©
             2002 Elsevier Science (USA).},
   Doi = {10.1006/tpbi.2001.1569},
   Key = {fds243499}
}

@article{fds243411,
   Author = {Calabrese, PP and Durrett, RT and Aquadro, CF},
   Title = {Dynamics of microsatellite divergence under stepwise
             mutation and proportional slippage/point mutation
             models.},
   Journal = {Genetics},
   Volume = {159},
   Number = {2},
   Pages = {839-852},
   Year = {2001},
   Month = {October},
   ISSN = {0016-6731},
   Abstract = {Recently Kruglyak, Durrett, Schug, and Aquadro showed that
             microsatellite equilibrium distributions can result from a
             balance between polymerase slippage and point mutations.
             Here, we introduce an elaboration of their model that keeps
             track of all parts of a perfect repeat and a simplification
             that ignores point mutations. We develop a detailed
             mathematical theory for these models that exhibits
             properties of microsatellite distributions, such as positive
             skewness of allele lengths, that are consistent with data
             but are inconsistent with the predictions of the stepwise
             mutation model. We use our theoretical results to analyze
             the successes and failures of the genetic distances
             (delta(mu))(2) and D(SW) when used to date four divergences:
             African vs. non-African human populations, humans vs.
             chimpanzees, Drosophila melanogaster vs. D. simulans, and
             sheep vs. cattle. The influence of point mutations explains
             some of the problems with the last two examples, as does the
             fact that these genetic distances have large stochastic
             variance. However, we find that these two features are not
             enough to explain the problems of dating the
             human-chimpanzee split. One possible explanation of this
             phenomenon is that long microsatellites have a mutational
             bias that favors contractions over expansions.},
   Key = {fds243411}
}

@article{fds243493,
   Author = {Arkendra, DE and Ferguson, M and Sindi, S and Durrett,
             R},
   Title = {The equilibrium distribution for a generalized
             Sankoff-Ferretti model accurately predicts chromosome size
             distributions in a wide variety of species},
   Journal = {Journal of Applied Probability},
   Volume = {38},
   Number = {2},
   Pages = {324-334},
   Publisher = {Cambridge University Press (CUP)},
   Year = {2001},
   Month = {June},
   ISSN = {0021-9002},
   url = {http://dx.doi.org/10.1239/jap/996986747},
   Abstract = {Sankoff and Ferretti (1996) introduced several models of the
             evolution of chromosome size by reciprocal translocations,
             where for simplicity they ignored the existence of
             centromeres. However, when they compared the models to data
             on six organisms they found that their short chromosomes
             were too short, and their long chromosomes were too long.
             Here, we consider a generalization of their proportional
             model with explicit chromosome centromeres and introduce
             fitness functions based on recombination probabilities and
             on the length of the longest chromosome arm. We find a
             simple formula for the stationary distribution for our model
             which fits the data on chromosome lengths in many, but not
             all, species.},
   Doi = {10.1239/jap/996986747},
   Key = {fds243493}
}

@article{fds243491,
   Author = {Durrett, R and Limic, V},
   Title = {On the quantity and quality of single nucleotide
             polymorphisms in the human genome},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {93},
   Number = {1},
   Pages = {1-24},
   Publisher = {Elsevier BV},
   Year = {2001},
   Month = {May},
   url = {http://dx.doi.org/10.1016/S0304-4149(00)00090-9},
   Abstract = {Single nucleotide polymorphisms (SNPs) are useful markers
             for locating genes since they occur throughout the human
             genome and thousands can be scored at once using DNA
             microarrays. Here, we use branching processes and coalescent
             theory to show that if one uses Kruglyak's (Nature Gen. 12
             (1999) 139-144) model of the growth of the human population
             and one assumes an average mutation rate of 1×10-8per
             nucleotide per generation then there are about 5.7 million
             SNP's in the human genome, or one every 526 base pairs. We
             also obtain results for the number of SNPs that will be
             found in samples of sizes n≥2 to gain insight into the
             number that will be found by various experimental
             procedures. © 2001 Elsevier Science B.V.},
   Doi = {10.1016/S0304-4149(00)00090-9},
   Key = {fds243491}
}

@article{fds243489,
   Author = {Diaconis, P and Durrett, R},
   Title = {Chutes and Ladders in Markov Chains},
   Journal = {Journal of Theoretical Probability},
   Volume = {14},
   Number = {3},
   Pages = {899-926},
   Year = {2001},
   url = {http://dx.doi.org/10.1023/A:1017509611178},
   Abstract = {We investigate how the stationary distribution of a Markov
             chain changes when transitions from a single state are
             modified. In particular, adding a single directed edge to
             nearest neighbor random walk on a finite discrete torus in
             dimensions one, two, or three changes the stationary
             distribution linearly, logarithmically, or only locally.
             Related results are derived for birth and death chains
             approximating Bessel diffusions and for random walk on the
             Sierpinski gasket.},
   Doi = {10.1023/A:1017509611178},
   Key = {fds243489}
}

@article{fds243490,
   Author = {Sundell, NM and Durrett, RT},
   Title = {Exponential distance statistics to detect the effects of
             population subdivision},
   Journal = {Theoretical Population Biology},
   Volume = {60},
   Number = {2},
   Pages = {107-116},
   Year = {2001},
   ISSN = {0040-5809},
   url = {http://dx.doi.org/10.1006/tpbi.2001.1522},
   Abstract = {Statistical tests are needed to determine whether spatial
             structure has had a significant effect on the genetic
             differentiation of subpopulations. Here we introduce a new
             family of statistics based on a sum of an exponential
             function of the distances between individuals, which can be
             used with any genetic distance (e.g., nucleotide
             differences, number of nonshared alleles, or separation on a
             phylogenetic tree). The power of the tests to detect genetic
             differentiation in Wright-Fisher island models and stepping
             stone models was calculated for various sample sizes, rates
             of migration and mutation, and definitions of spatial
             neighborhoods. We found that our new test was in some cases
             more powerful than the K S* statistic of Hudson et al. (Mol.
             Biol. Evol. 9, 138-151, 1992), but in all cases was slightly
             less powerful than both a traditional X 2 test without
             lumping of rare haplotypes and the S nn test of Hudson
             (Genetics 155, 2011-2014, 2000). However, when we applied
             our new tests to three data sets, we found in some cases
             highly significant results that were missed by the other
             tests. © 2001 Academic Press.},
   Doi = {10.1006/tpbi.2001.1522},
   Key = {fds243490}
}

@article{fds243481,
   Author = {Kruglyak, S and Durrett, R and Schug, MD and Aquadro,
             CF},
   Title = {Distribution and abundance of microsatellites in the yeast
             genome can Be explained by a balance between slippage events
             and point mutations.},
   Journal = {Molecular Biology and Evolution},
   Volume = {17},
   Number = {8},
   Pages = {1210-1219},
   Year = {2000},
   Month = {August},
   ISSN = {0737-4038},
   url = {http://dx.doi.org/10.1093/oxfordjournals.molbev.a026404},
   Abstract = {We fit a Markov chain model of microsatellite evolution
             introduced by Kruglyak et al. to data on all di-, tri-, and
             tetranucleotide repeats in the yeast genome. Our results
             suggest that many features of the distribution of abundance
             and length of microsatellites can be explained by this
             simple model, which incorporates a competition between
             slippage events and base pair substitutions, with no need to
             invoke selection or constraints on the lengths. Our results
             provide some new information on slippage rates for
             individual repeat motifs, which suggest that AT-rich
             trinucleotide repeats have higher slippage rates. As our
             model predicts, we found that many repeats were adjacent to
             shorter repeats of the same motif. However, we also found a
             significant tendency of microsatellites of different motifs
             to cluster.},
   Doi = {10.1093/oxfordjournals.molbev.a026404},
   Key = {fds243481}
}

@article{fds243485,
   Author = {Vision, TJ and Brown, DG and Shmoys, DB and Durrett, RT and Tanksley,
             SD},
   Title = {Selective mapping: a strategy for optimizing the
             construction of high-density linkage maps.},
   Journal = {Genetics},
   Volume = {155},
   Number = {1},
   Pages = {407-420},
   Year = {2000},
   Month = {May},
   Abstract = {Historically, linkage mapping populations have consisted of
             large, randomly selected samples of progeny from a given
             pedigree or cell lines from a panel of radiation hybrids. We
             demonstrate that, to construct a map with high genome-wide
             marker density, it is neither necessary nor desirable to
             genotype all markers in every individual of a large mapping
             population. Instead, a reduced sample of individuals bearing
             complementary recombinational or radiation-induced
             breakpoints may be selected for genotyping subsequent
             markers from a large, but sparsely genotyped, mapping
             population. Choosing such a sample can be reduced to a
             discrete stochastic optimization problem for which the goal
             is a sample with breakpoints spaced evenly throughout the
             genome. We have developed several different methods for
             selecting such samples and have evaluated their performance
             on simulated and actual mapping populations, including the
             Lister and Dean Arabidopsis thaliana recombinant inbred
             population and the GeneBridge 4 human radiation hybrid
             panel. Our methods quickly and consistently find
             much-reduced samples with map resolution approaching that of
             the larger populations from which they are derived. This
             approach, which we have termed selective mapping, can
             facilitate the production of high-quality, high-density
             genome-wide linkage maps.},
   Key = {fds243485}
}

@article{fds243484,
   Author = {Liu, YC and Durrett, R and Milgroom, MG},
   Title = {A spatially-structured stochastic model to simulate
             heterogenous transmission of viruses in fungal
             populations},
   Journal = {Ecological Modelling},
   Volume = {127},
   Number = {2-3},
   Pages = {291-308},
   Publisher = {Elsevier BV},
   Year = {2000},
   Month = {March},
   url = {http://dx.doi.org/10.1016/S0304-3800(99)00216-1},
   Abstract = {A spatially explicit, interacting particle system model was
             developed to simulate the heterogeneous transmission of
             viruses in fungal populations. This model is based primarily
             on hypoviruses in the chestnut blight fungus, Cryphonectria
             parasitica, which debilitate their hosts and function as
             biological control agents. An important characteristic of
             this system is that virus transmission occurs freely between
             individuals in the same genetically defined vegetative
             compatibility (vc) type, but is restricted among individuals
             in different vc types, resulting in heterogeneous
             transmission. An additional source of heterogeneity is
             spatial structure in host populations; viruses are dispersed
             by fungal spores which disperse relatively short distances.
             The model showed that vc type diversity is highly correlated
             to the horizontal transmission rate and therefore
             significantly affects virus invasion. The probability of
             virus invasion decreased as the diversity of vc types
             increased. We also demonstrated that virus transmission
             would be overestimated if we assumed virus transmission was
             homogeneous, ignoring both genetic and spatial
             heterogeneity. Genetic and spatial heterogeneity are not
             independent because both are affected by the reproductive
             biology of the fungus. In asexual populations, restricted
             fungus dispersal resulted in nonrandom spatial patterns of
             vc types, increasing the chance of contact between
             vegetatively compatible individuals, and promoting virus
             transmission. In contrast, virus transmission was poor in
             sexual populations due to spatial randomization of vc types
             by long distance dispersed sexual spores. Finally, this
             model was used to evaluate the release of genetically
             engineered virus-infected strains for disease management.
             The release of transgenic strains resulted in only
             marginally greater virus establishment than for
             non-transgenic strains. Virus invasion was still restricted
             by vc type diversity in the resident fungus population.
             Simulation of inundative releases of transgenic
             virus-infected strains slightly improved virus
             establishment, but viruses did not persist after treatment
             was terminated. (C) 2000 Elsevier Science
             B.V.},
   Doi = {10.1016/S0304-3800(99)00216-1},
   Key = {fds243484}
}

@article{fds243483,
   Author = {Cox, JT and Durrett, R and Perkins, EA},
   Title = {Rescaled voter models converge to super-Brownian
             motion},
   Journal = {The Annals of Probability},
   Volume = {28},
   Number = {1},
   Pages = {185-234},
   Publisher = {Institute of Mathematical Statistics},
   Year = {2000},
   Month = {January},
   url = {http://dx.doi.org/10.1214/aop/1019160117},
   Abstract = {We show that a sequence of voter models, suitably rescaled
             in space and time, converges weakly to super-Brownian
             motion. The result includes both nearest neighbor and longer
             range voter models and complements a limit theorem of
             Mueller and Tribe in one dimension.},
   Doi = {10.1214/aop/1019160117},
   Key = {fds243483}
}

@article{fds243482,
   Author = {Durrett, R and Buttel, L and Harrison, R},
   Title = {Spatial models for hybrid zones},
   Journal = {Heredity},
   Volume = {84},
   Number = {1},
   Pages = {9-19},
   Year = {2000},
   ISSN = {0018-067X},
   url = {http://dx.doi.org/10.1046/j.1365-2540.2000.00566.x},
   Abstract = {We introduce a spatially explicit model of natural hybrid
             zones that allows us to consider how patterns of allele
             frequencies and linkage disequilibria change over time. We
             examine the influence of hybrid zone origins on patterns of
             variation at two loci, a locus under selection in a
             two-patch environment, and a linked neutral locus. We
             consider several possible starting conditions that represent
             explicit realizations of two alternative scenarios for
             hybrid zone origins: primary intergradation and secondary
             contact. Our results indicate that in some circumstances,
             differences in hybrid zone origins will result in
             substantially different patterns of variation that may
             persist for thousands of generations. Our conclusions are
             generally similar to those previously derived from partial
             differential equations, but there are also some important
             differences.},
   Doi = {10.1046/j.1365-2540.2000.00566.x},
   Key = {fds243482}
}

@article{fds243486,
   Author = {Durrett, R and Schinazi, RB},
   Title = {Boundary Modified Contact Processes},
   Journal = {Journal of Theoretical Probability},
   Volume = {13},
   Number = {2},
   Pages = {575-594},
   Year = {2000},
   Abstract = {We introduce a one dimensional contact process for which
             births to the right of the rightmost particle and to the
             left of the leftmost particle occur at rate λe (where e is
             for external). Other births occur at rate λi (where i is
             for internal). Deaths occur at rate 1. The case λe = λi is
             the well known basic contact process for which there is a
             critical value λc &gt; 1 such that if the birth rate is
             larger than λc the process has a positive probability of
             surviving. Our main motivation here is to understand the
             relative importance of the external birth rates. We show
             that if λe ≤ 1 then the process always dies out while if
             λe &gt; 1 and if λi is large enough then the process may
             survive. We also show that if λi &lt; λc the process dies
             out for all λe. To extend this notion to d &gt; 1 we
             introduce a second process that has an epidemiological
             interpretation. For this process each site can be in one of
             three states: infected, a susceptible that has never been
             infected, or a susceptible that has been infected
             previously. Furthermore, the rates at which the two types of
             susceptible become infected are different. We obtain some
             information about the phase diagram about this case as
             well.},
   Key = {fds243486}
}

@article{fds243487,
   Author = {Durrett, R and Levin, S},
   Title = {Lessons on pattern formation from planet
             WATOR},
   Journal = {Journal of Theoretical Biology},
   Volume = {205},
   Number = {2},
   Pages = {201-214},
   Year = {2000},
   ISSN = {0022-5193},
   url = {http://dx.doi.org/10.1006/jtbi.2000.2061},
   Abstract = {It is well known that if reacting species experience unequal
             diffusion rates, then dynamics that lead to a constant
             steady state in a 'well-mixed' environment can in a spatial
             setting lead to interesting patterns. In this paper, we
             focus on complementary pattern formation mechanisms that
             operate even when the diffusion rates are equal. In
             particular, we can say that when the mean-field ODE has an
             attracting periodic orbit then the stochastic spatial model
             will have large-scale spatial structures in equilibrium. We
             explore this mechanism in depth through the dynamics of the
             simulator WATOR. (C) 2000 Academic Press.},
   Doi = {10.1006/jtbi.2000.2061},
   Key = {fds243487}
}

@article{fds243488,
   Author = {Broughton, RE and Stanley, SE and Durrett, RT},
   Title = {Quantification of homoplasy for nucleotide transitions and
             transversions and a reexamination of assumptions in weighted
             phylogenetic analysis},
   Journal = {Systematic Biology},
   Volume = {49},
   Number = {4},
   Pages = {617-627},
   Year = {2000},
   Abstract = {Nucleotide transitions are frequently down-weighted relative
             to transversions in phylogenetic analysis. This is based on
             the assumption that transitions, by virtue of their greater
             evolutionary rate, exhibit relatively more homoplasy and are
             therefore less reliable phylogenetic characters. Relative
             amounts of homoplastic and consistent transition and
             transversion changes in mitochondrial protein coding genes
             were determined from character-state reconstructions on a
             highly corroborated phylogeny of mammals. We found that
             although homoplasy was related to evolutionary rates and was
             greater for transitions, the absolute number of consistent
             transitions greatly exceeded the number of consistent
             transversions. Consequently, transitions provided
             substantially more useful phylogenetic information than
             transversions. These results suggest that down-weighting
             transitions may be unwarranted in many cases. This
             conclusion was supported by the fact that a range of
             transition: transversion weighting schemes applied to
             various mitochondrial genes and genomic partitions rarely
             provided improvement in phylogenetic estimates relative to
             equal weighting, and in some cases weighting transitions
             more heavily than transversions was most
             effective.},
   Key = {fds243488}
}

@article{fds243477,
   Author = {Durrett, R and Perkins, EA},
   Title = {Rescaled contact processes converge to super-Brownian motion
             in two or more dimensions},
   Journal = {Probability Theory and Related Fields},
   Volume = {114},
   Number = {3},
   Pages = {309-399},
   Publisher = {Springer Nature},
   Year = {1999},
   Month = {June},
   url = {http://dx.doi.org/10.1007/s004400050228},
   Abstract = {We show that in dimensions two or more a sequence of long
             range contact processes suitably rescaled in space and time
             converges to a super-Brownian motion with drift. As a
             consequence of this result we can improve the results of
             Bramson, Durrett, and Swindle (1989) by replacing their
             order of magnitude estimates of how close the critical value
             is to 1 with sharp asymptotics.},
   Doi = {10.1007/s004400050228},
   Key = {fds243477}
}

@article{fds243478,
   Author = {Molofsky, J and Durrett, R and Dushoff, J and Griffeath, D and Levin,
             S},
   Title = {Local frequency dependence and global coexistence.},
   Journal = {Theoretical Population Biology},
   Volume = {55},
   Number = {3},
   Pages = {270-282},
   Year = {1999},
   Month = {June},
   ISSN = {0040-5809},
   url = {http://dx.doi.org/10.1006/tpbi.1998.1404},
   Abstract = {In sessile organisms such as plants, interactions occur
             locally so that important ecological aspects like frequency
             dependence are manifest within local neighborhoods. Using
             probabilistic cellular automata models, we investigated how
             local frequency-dependent competition influenced whether two
             species could coexist. Individuals of the two species were
             randomly placed on a grid and allowed to interact according
             to local frequency-dependent rules. For four different
             frequency-dependent scenarios, the results indicated that
             over a broad parameter range the two species could coexist.
             Comparisons between explicit spatial simulations and the
             mean-field approximation indicate that coexistence occurs
             over a broader region in the explicit spatial
             simulation.},
   Doi = {10.1006/tpbi.1998.1404},
   Key = {fds243478}
}

@article{fds243476,
   Author = {Durrett, R},
   Title = {Stochastic spatial models},
   Journal = {Siam Review},
   Volume = {41},
   Number = {4},
   Pages = {677-718},
   Publisher = {Society for Industrial & Applied Mathematics
             (SIAM)},
   Year = {1999},
   Month = {January},
   url = {http://dx.doi.org/10.1137/S0036144599354707},
   Abstract = {In the models we will consider, space is represented by a
             grid of sites that can be in one of a finite number of
             states and that change at rates that depend on the states of
             a finite number of sites. Our main aim here is to explain an
             idea of Durrett and Levin (1994): the behavior of these
             models can be predicted from the properties of the mean
             field ODE, i.e., the equations for the densities of the
             various types that result from pretending that all sites are
             always independent. We will illustrate this picture through
             a discussion of eight families of examples from statistical
             mechanics, genetics, population biology, epidemiology, and
             ecology. Some of our findings are only conjectures based on
             simulation, but in a number of cases we are able to prove
             results for systems with `fast stirring' by exploiting
             connections between the spatial model and an associated
             reaction diffusion equation.},
   Doi = {10.1137/S0036144599354707},
   Key = {fds243476}
}

@article{fds243479,
   Author = {Durrett, R and Granovsky, BL and Gueron, S},
   Title = {The Equilibrium Behavior of Reversible Coagulation-Fragmentation
             Processes},
   Journal = {Journal of Theoretical Probability},
   Volume = {12},
   Number = {2},
   Pages = {447-474},
   Year = {1999},
   Month = {January},
   url = {http://dx.doi.org/10.1023/A:1021682212351},
   Abstract = {The coagulation-fragmentation process models the stochastic
             evolution of a population of N particles distributed into
             groups of different sizes that coagulate and fragment at
             given rates. The process arises in a variety of contexts and
             has been intensively studied for a long time. As a result,
             different approximations to the model were suggested. Our
             paper deals with the exact model which is viewed as a
             time-homogeneous interacting particle system on the state
             space ΩN, the set of all partitions of N. We obtain the
             stationary distribution (invariant measure) on ΩN for the
             whole class of reversible coagulation-fragmentation
             processes, and derive explicit expressions for important
             functionals of this measure, in particular, the expected
             numbers of groups of all sizes at the steady state. We also
             establish a characterization of the transition rates that
             guarantee the reversibility of the process. Finally, we make
             a comparative study of our exact solution and the
             approximation given by the steady-state solution of the
             coagulation-fragmentation integral equation, which is known
             in the literature. We show that in some cases the latter
             approximation can considerably deviate from the exact
             solution.},
   Doi = {10.1023/A:1021682212351},
   Key = {fds243479}
}

@article{fds243480,
   Author = {Durrett, R and Kruglyak, S},
   Title = {A new stochastic model of microsatellite
             evolution},
   Journal = {Journal of Applied Probability},
   Volume = {36},
   Number = {3},
   Pages = {621-631},
   Publisher = {Cambridge University Press (CUP)},
   Year = {1999},
   Month = {January},
   url = {http://dx.doi.org/10.1017/S0021900200017447},
   Abstract = {We introduce a continuous-time Markov chain model for the
             evolution of microsatellites, simple sequence repeats in
             DNA. We prove the existence of a unique stationary
             distribution for our model, and fit the model to data from
             approximately 106 base pairs of DNA from fruit flies, mice,
             and humans. The slippage rates from the best fit for our
             model are consistent with experimental findings.},
   Doi = {10.1017/S0021900200017447},
   Key = {fds243480}
}

@article{fds243475,
   Author = {Kruglyak, S and Durrett, RT and Schug, MD and Aquadro,
             CF},
   Title = {Equilibrium distributions of microsatellite repeat length
             resulting from a balance between slippage events and point
             mutations.},
   Journal = {Proceedings of the National Academy of Sciences of the
             United States of America},
   Volume = {95},
   Number = {18},
   Pages = {10774-10778},
   Year = {1998},
   Month = {September},
   ISSN = {0027-8424},
   url = {http://dx.doi.org/10.1073/pnas.95.18.10774},
   Abstract = {We describe and test a Markov chain model of microsatellite
             evolution that can explain the different distributions of
             microsatellite lengths across different organisms and repeat
             motifs. Two key features of this model are the dependence of
             mutation rates on microsatellite length and a mutation
             process that includes both strand slippage and point
             mutation events. We compute the stationary distribution of
             allele lengths under this model and use it to fit DNA data
             for di-, tri-, and tetranucleotide repeats in humans, mice,
             fruit flies, and yeast. The best fit results lead to
             slippage rate estimates that are highest in mice, followed
             by humans, then yeast, and then fruit flies. Within each
             organism, the estimates are highest in di-, then tri-, and
             then tetranucleotide repeats. Our estimates are consistent
             with experimentally determined mutation rates from other
             studies. The results suggest that the different length
             distributions among organisms and repeat motifs can be
             explained by a simple difference in slippage rates and that
             selective constraints on length need not be
             imposed.},
   Doi = {10.1073/pnas.95.18.10774},
   Key = {fds243475}
}

@article{fds243470,
   Author = {Durrett, R and Levin, S},
   Title = {ERRATUM},
   Journal = {Theoretical Population Biology},
   Volume = {53},
   Number = {3},
   Pages = {284-284},
   Publisher = {Elsevier BV},
   Year = {1998},
   Month = {June},
   url = {http://dx.doi.org/10.1006/tpbi.1998.1374},
   Doi = {10.1006/tpbi.1998.1374},
   Key = {fds243470}
}

@article{fds243473,
   Author = {Bramson, M and Cox, JT and Durrett, R},
   Title = {A spatial model for the abundance of species},
   Journal = {The Annals of Probability},
   Volume = {26},
   Number = {2},
   Pages = {658-709},
   Publisher = {Institute of Mathematical Statistics},
   Year = {1998},
   Month = {April},
   url = {http://dx.doi.org/10.1214/aop/1022855647},
   Abstract = {The voter model, with mutations occurring at a positive rate
             a, has a unique equilibrium distribution. We investigate the
             logarithms of the relative abundance of species for these
             distributions in d ≥ 2. We show that, as α → 0, the
             limiting distribution is right triangular in d = 2 and
             uniform in d > 3. We also obtain more detailed results for
             the histograms that biologists use to estimate the
             underlying density functions.},
   Doi = {10.1214/aop/1022855647},
   Key = {fds243473}
}

@article{fds243474,
   Author = {Durrett, R and Levin, S},
   Title = {Spatial aspects of interspecific competition},
   Journal = {Theoretical Population Biology},
   Volume = {53},
   Number = {1},
   Pages = {30-43},
   Year = {1998},
   ISSN = {0040-5809},
   url = {http://dx.doi.org/10.1006/tpbi.1997.1338},
   Abstract = {Using several variants of a stochastic spatial model
             introduced by Silvertown et al., we investigate the effect
             of spatial distribution of individuals on the outcome of
             competition. First, we prove rigorously that if one species
             has a competitive advantage over each of the others, then
             eventually it takes over all the sites in the system.
             Second, we examine tradeoffs between competition and
             dispersal distance in a two-species system. Third, we
             consider a cyclic competitive relationship between three
             types. In this case, a nonspatial treatment leads to
             densities that follow neutrally stable cycles or even
             unstable spiral solutions, while a spatial model yields a
             stationary distribution with an interesting spatial
             structure.},
   Doi = {10.1006/tpbi.1997.1338},
   Key = {fds243474}
}

@article{fds243471,
   Author = {Durrett, R and Levin, S},
   Title = {Allelopathy in Spatially Distributed Populations},
   Journal = {Journal of Theoretical Biology},
   Volume = {185},
   Number = {2},
   Pages = {165-171},
   Publisher = {Elsevier BV},
   Year = {1997},
   Month = {March},
   url = {http://dx.doi.org/10.1006/jtbi.1996.0292},
   Abstract = {In a homogeneously mixing population of E. coli,
             colicin-producing and colicin-sensitive strategies both may
             be evolutionarily stable for certain parameter ranges, with
             the outcome of competition determined by initial conditions.
             In contrast, in a spatially-structured population, there is
             a unique ESS for any given set of parameters; the outcome is
             determined by how effective allelopathy is in relation to
             its costs. Furthermore, in a spatially-structured
             environment, a dynamic equilibrium may be sustained among a
             colicin-sensitive type, a high colicin-producing type, and a
             'cheater' that expends less on colicin production but is
             resistant.},
   Doi = {10.1006/jtbi.1996.0292},
   Key = {fds243471}
}

@article{fds243469,
   Author = {Durrett, R and Neuhauser, C},
   Title = {Coexistence results for some competition
             models},
   Journal = {The Annals of Applied Probability},
   Volume = {7},
   Number = {1},
   Pages = {10-45},
   Publisher = {Institute of Mathematical Statistics},
   Year = {1997},
   Month = {February},
   url = {http://dx.doi.org/10.1214/aoap/1034625251},
   Abstract = {Barley yellow dwarf is a widespread disease that affects
             small grains and many grass species, as well as wheat,
             barley and oat. The disease is caused by an aphid
             transmitted virus. Rochow conducted a study near Ithaca, New
             York, which showed that a shift in the dominant strain
             occurred between 1957 and 1976. Motivated by this
             phenomenon, we develop a model for the competition between
             different strains of the barley yellow dwarf virus. Our main
             goal is to understand the phase diagram of the model, that
             is, to identify parameter values where one strain
             competitively excludes the other strain and where both
             strains coexist. Our analysis applies to a number of other
             systems as well, for example to a model of competition of
             water flea species studied by Hanski and Ranta and
             Bengtsson.},
   Doi = {10.1214/aoap/1034625251},
   Key = {fds243469}
}

@article{fds243472,
   Author = {Chen, ZQ and Durrett, R and Ma, G},
   Title = {Holomorphic diffusions and boundary behavior of harmonic
             functions},
   Journal = {The Annals of Probability},
   Volume = {25},
   Number = {3},
   Pages = {1103-1134},
   Publisher = {Institute of Mathematical Statistics},
   Year = {1997},
   Month = {January},
   url = {http://dx.doi.org/10.1214/aop/1024404507},
   Abstract = {We study a family of differential operators {Lα, α ≥ 0}
             in the unit ball D of Cn with n ≥ 2 that generalize the
             classical Laplacian, α = 0, and the conformal Laplacian, α
             = 1/2 (that is, the Laplace-Beltrami operator for Bergman
             metric in D). Using the diffusion processes associated with
             these (degenerate) differential operators, the boundary
             behavior of Lα-harmonic functions is studied in a unified
             way for 0 ≤ α ≤ 1/2. More specifically, we show that a
             bounded Lα-harmonic function in D has boundary limits in
             approaching regions at almost every boundary point and the
             boundary approaching region increases from the Stolz cone to
             the Korányi admissible region as α runs from 0 to 1/2. A
             local version for this Fatou-type result is also
             established.},
   Doi = {10.1214/aop/1024404507},
   Key = {fds243472}
}

@article{fds243468,
   Author = {Allouba, H and Durrett, R and Hawkes, J and Perkins,
             E},
   Title = {Super-Tree Random Measures},
   Journal = {Journal of Theoretical Probability},
   Volume = {10},
   Number = {3},
   Pages = {773-794},
   Year = {1997},
   Abstract = {We use supercritical branching processes with random walk
             steps of geometrically decreasing size to construct random
             measures. Special cases of our construction give close
             relatives of the super-(spherically symmetric stable)
             processes. However, other cases can produce measures with
             very smooth densities in any dimension.},
   Key = {fds243468}
}

@article{fds243466,
   Author = {Durrett, R and Levin, S},
   Title = {Spatial Models for Species-Area Curves},
   Journal = {Journal of Theoretical Biology},
   Volume = {179},
   Number = {2},
   Pages = {119-127},
   Publisher = {Elsevier BV},
   Year = {1996},
   Month = {March},
   url = {http://dx.doi.org/10.1006/jtbi.1996.0053},
   Abstract = {Inspired by earlier work of Hubbell, we introduce a simple
             spatial model to explain observed species-area curves. As in
             the theory of MacArthur and Wilson, our curves result from a
             balance between migration and extinction. Our model predicts
             that the wide range of slopes of species-area curves is due
             to the differences in the rates at which new species enter
             this system. However, two other predictions, that the slope
             increases with increasing migration/mutation and that the
             curves for remote islands are flatter than those for near
             islands, are at odds with some interpretations of data. This
             suggests either that the data have been misinterpreted, or
             that the model is not sufficient to explain
             them.},
   Doi = {10.1006/jtbi.1996.0053},
   Key = {fds243466}
}

@article{fds243467,
   Author = {Bramson, M and Cox, JT and Durrett, R},
   Title = {Spatial models for species area curves},
   Journal = {The Annals of Probability},
   Volume = {24},
   Number = {4},
   Pages = {1727-1751},
   Publisher = {Institute of Mathematical Statistics},
   Year = {1996},
   url = {http://dx.doi.org/10.1214/aop/1041903204},
   Abstract = {The relationship between species number and area is an old
             problem in biology. We propose here an interacting particle
             system - the multitype voter model with mutation - as a
             mathematical model to study this problem. We analyze the
             species area curves of this model as the mutation rate α
             tends to zero. We obtain two basic types of behavior
             depending on the size of the spatial region under
             consideration. If the region is a square with area α-r, r
             &gt; 1, then, for small α, the number of species is of
             order α1r(log α)2, whereas if r &lt; 1, the number of
             species is bounded.},
   Doi = {10.1214/aop/1041903204},
   Key = {fds243467}
}

@article{fds323656,
   Author = {Cox, JT and Durrett, R},
   Title = {Hybrid zones and voter model interfaces},
   Journal = {Bernoulli},
   Volume = {1},
   Number = {4},
   Pages = {343-370},
   Publisher = {Bernoulli Society for Mathematical Statistics and
             Probability},
   Year = {1995},
   Month = {December},
   url = {http://dx.doi.org/10.3150/bj/1193758711},
   Abstract = {We study the dynamics of hybrid zones in the absence of
             selection. In dimensions d > 1 the width of the hybrid zone
             grows as √t but in one dimension the width converges to a
             non-degenerate limit. We believe that tight interfaces are
             common in one-dimensional particle systems. © 1995 Chapman
             & Hall.},
   Doi = {10.3150/bj/1193758711},
   Key = {fds323656}
}

@article{fds243463,
   Author = {Durrett, R and Swindle, G},
   Title = {Coexistence results for catalysts},
   Journal = {Probability Theory and Related Fields},
   Volume = {98},
   Number = {4},
   Pages = {489-515},
   Publisher = {Springer Nature},
   Year = {1994},
   Month = {December},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/bf01192836},
   Abstract = {In this paper we consider a modification of Ziff, Gulari and
             Barshad's (1986) model of oxidation of carbon monoxide on a
             catalyst surface in which the reactants are mobile on the
             catalyst surface. We find regions in the parameter space in
             which poisoning occurs (the catalyst surface becomes
             completely occupied by one type of atom) and another in
             which there is a translation invariant stationary
             distribution in which the two atoms have positive density.
             The last result is proved by exploiting a connection between
             the particle system with fast stirring and a limiting system
             of reaction diffusion equations. © 1994
             Springer-Verlag.},
   Doi = {10.1007/bf01192836},
   Key = {fds243463}
}

@article{fds243464,
   Author = {Durrett, R and Levin, S},
   Title = {The importance of being discrete (and spatial)},
   Journal = {Theoretical Population Biology},
   Volume = {46},
   Number = {3},
   Pages = {363-394},
   Publisher = {Elsevier BV},
   Year = {1994},
   Month = {January},
   url = {http://dx.doi.org/10.1006/tpbi.1994.1032},
   Abstract = {We consider and compare four approaches to modeling the
             dynamics of spatially distributed systems: mean field
             approaches (described by ordinary differential equations) in
             which every individual is considered to have equal
             probability of interacting with every other individual;
             patch models that group discrete individuals into patches
             without additional spatial structure; reaction-diffusion
             equations, in which infinitesimal individuals are
             distributed in space; and interacting particle systems, in
             which individuals are discrete and space is treated
             explicitly. We apply these four approaches to three examples
             of species interactions in spatially distributed populations
             and compare their predictions. Each represents different
             assumptions about the biology and hence a comparison among
             them has biological as well as modeling implications. In the
             first case all four approaches agree, in the second the
             spatial models disagree with the nonspatial ones, while in
             the third the stochastic models with discrete individuals
             disagree with the ones based on differential equations. We
             show further that the limiting reaction-diffusion equations
             associated with particle systems can have different
             qualitative behavior from those obtained by simply adding
             diffusion terms to mean field equations. © 1994 Academic
             Press. All rights reserved.},
   Doi = {10.1006/tpbi.1994.1032},
   Key = {fds243464}
}

@article{fds243465,
   Author = {Durrett, R and Levin, SA},
   Title = {Stochastic spatial models: A user's guide to ecological
             applications},
   Journal = {Philosophical Transactions of the Royal Society of London.
             Series B, Biological Sciences},
   Volume = {343},
   Number = {1305},
   Pages = {329-350},
   Publisher = {The Royal Society},
   Year = {1994},
   Month = {January},
   ISSN = {0962-8436},
   url = {http://dx.doi.org/10.1098/rstb.1994.0028},
   Doi = {10.1098/rstb.1994.0028},
   Key = {fds243465}
}

@article{fds323657,
   Author = {Durrett, R and Griffeath, D},
   Title = {Asymptotic Behavior of Excitable Cellular
             Automata},
   Journal = {Experimental Mathematics},
   Volume = {2},
   Number = {3},
   Pages = {183-208},
   Publisher = {Informa UK Limited},
   Year = {1993},
   Month = {January},
   url = {http://dx.doi.org/10.1080/10586458.1993.10504277},
   Doi = {10.1080/10586458.1993.10504277},
   Key = {fds323657}
}

@article{fds243462,
   Author = {Durrett, RT and Rogers, LCG},
   Title = {Asymptotic behavior of Brownian polymers},
   Journal = {Probability Theory and Related Fields},
   Volume = {92},
   Number = {3},
   Pages = {337-349},
   Publisher = {Springer Nature},
   Year = {1992},
   Month = {September},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/BF01300560},
   Abstract = {We consider a system that models the shape of a growing
             polymer. Our basic problem concerns the asymptotic behavior
             of Xt, the location of the end of the polymer at time t. We
             obtain bounds on Xt in the (physically uninteresting) case
             that d=1 and the interaction function f(x)≥0. If, in
             addition, f(x) behaves for large x like Cx-β with β<1 we
             obtain a strong law that gives the exact growth rate. ©
             1992 Springer-Verlag.},
   Doi = {10.1007/BF01300560},
   Key = {fds243462}
}

@article{fds243460,
   Author = {Durrett, R},
   Title = {Multicolor particle systems with large threshold and
             range},
   Journal = {Journal of Theoretical Probability},
   Volume = {5},
   Number = {1},
   Pages = {127-152},
   Publisher = {Springer Nature},
   Year = {1992},
   Month = {January},
   ISSN = {0894-9840},
   url = {http://dx.doi.org/10.1007/bf01046781},
   Abstract = {In this paper we consider the Greenberg-Hastings and cyclic
             color models. These models exhibit (at least) three
             different types of behavior. Depending on the number of
             colors and the size of two parameters called the threshold
             and range, the Greenberg-Hastings model either dies out, or
             has equilibria that consist of "debris" or "fire fronts".
             The phase diagram for the cyclic color models is more
             complicated. The main result of this paper, Theorem 1,
             proves that the debris phase exists for both systems. ©
             1992 Plenum Publishing Corporation.},
   Doi = {10.1007/bf01046781},
   Key = {fds243460}
}

@article{fds243461,
   Author = {Durrett, R and Steif, JE},
   Title = {Some rigorous results for the Greenberg-Hastings
             Model},
   Journal = {Journal of Theoretical Probability},
   Volume = {4},
   Number = {4},
   Pages = {669-690},
   Publisher = {Springer Nature},
   Year = {1991},
   Month = {October},
   ISSN = {0894-9840},
   url = {http://dx.doi.org/10.1007/bf01259549},
   Abstract = {In this paper, we obtain some rigorous results for a
             cellular automaton known as the Greenberg-Hastings Model.
             The state space is {0, 1, 2}Zd. The dynamics are
             deterministic and discrete time. A site which is 1 changes
             to 2, a site which is 2 changes to 0, and a site which is 0
             changes to a 1 if one of its 2 d neighbors is a 1. In one
             dimension, we compute the exact asymptotic rate at which the
             system dies out when started at random and compute the
             topological entropy. In two or more dimensions we show that
             starting from a nontrivial product measure, the limit exists
             as 3 m→∞ and is Bernoulli shift. Finally, we investigate
             the behavior of the system on a large finite box. © 1991
             Plenum Publishing Corporation.},
   Doi = {10.1007/bf01259549},
   Key = {fds243461}
}

@article{fds243457,
   Author = {Cox, JT and Durrett, R and Schinazi, R},
   Title = {The critical contact process seen from the right
             edge},
   Journal = {Probability Theory and Related Fields},
   Volume = {87},
   Number = {3},
   Pages = {325-332},
   Publisher = {Springer Nature},
   Year = {1991},
   Month = {September},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/bf01312213},
   Abstract = {Durrett (1984) proved the existence of an invariant measure
             for the critical and supercritical contact process seen from
             the right edge. Galves and Presutti (1987) proved, in the
             supercritical case, that the invariant measure was unique,
             and convergence to it held starting in any semi-infinite
             initial state. We prove the same for the critical contact
             process. We also prove that the process starting with one
             particle, conditioned to survive until time t, converges to
             the unique invariant measure as t→∞. © 1991
             Springer-Verlag.},
   Doi = {10.1007/bf01312213},
   Key = {fds243457}
}

@article{fds243459,
   Author = {Durrett, R and M�ller, AM},
   Title = {Complete convergence theorem for a competition
             model},
   Journal = {Probability Theory and Related Fields},
   Volume = {88},
   Number = {1},
   Pages = {121-136},
   Publisher = {Springer Nature},
   Year = {1991},
   Month = {March},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/bf01193585},
   Abstract = {In this paper we consider a hierarchical competition model.
             Durrett and Swindle have given sufficient conditions for the
             existence of a nontrivial stationary distribution. Here we
             show that under a slightly stronger condition, the complete
             convergence theorem holds and hence there is a unique
             nontrivial stationary distribution. © 1991
             Springer-Verlag.},
   Doi = {10.1007/bf01193585},
   Key = {fds243459}
}

@article{fds243455,
   Author = {Durrett, R and Swindle, G},
   Title = {Are there bushes in a forest?},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {37},
   Number = {1},
   Pages = {19-31},
   Publisher = {Elsevier BV},
   Year = {1991},
   Month = {January},
   ISSN = {0304-4149},
   url = {http://dx.doi.org/10.1016/0304-4149(91)90057-J},
   Abstract = {In this paper we consider a process in which each site x
             ε{lunate} Zd can be occupied by grass, bushes or trees and
             ask the question: Are there equilibria in which bushes and
             trees are both present? The answer is sometimes yes and
             sometimes no. © 1991.},
   Doi = {10.1016/0304-4149(91)90057-J},
   Key = {fds243455}
}

@article{fds243456,
   Author = {Durrett, R and Kesten, H and Waymire, E},
   Title = {On weighted heights of random trees},
   Journal = {Journal of Theoretical Probability},
   Volume = {4},
   Number = {1},
   Pages = {223-237},
   Publisher = {Springer Nature},
   Year = {1991},
   Month = {January},
   ISSN = {0894-9840},
   url = {http://dx.doi.org/10.1007/bf01047004},
   Abstract = {Consider the family tree T of a branching process starting
             from a single progenitor and conditioned to have v=v(T)
             edges (total progeny). To each edge &lt;e&gt; we associate a
             weight W(e). The weights are i.i.d. random variables and
             independent of T. The weighted height of a self-avoiding
             path in T starting at the root is the sum of the weights
             associated with the path. We are interested in the
             asymptotic distribution of the maximum weighted path height
             in the limit as v=n→∞. Depending on the tail of the
             weight distribution, we obtain the limit in three cases. In
             particular if y2P(W(e)&gt; y)→0, then the limit
             distribution depends strongly on the tree and, in fact, is
             the distribution of the maximum of a Brownian excursion. If
             the tail of the weight distribution is regularly varying
             with exponent 0≤α&lt;2, then the weight swamps the tree
             and the answer is the asymptotic distribution of the maximum
             edge weight in the tree. There is a borderline case, namely,
             P(W(e)&gt; y)∼cy-2 as y→∞, in which the limit
             distribution exists but involves both the tree and the
             weights in a more complicated way. © 1991 Plenum Publishing
             Corporation.},
   Doi = {10.1007/bf01047004},
   Key = {fds243456}
}

@article{fds243458,
   Author = {Bramson, M and Wan-ding, D and Durrett, R},
   Title = {Annihilating branching processes},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {37},
   Number = {1},
   Pages = {1-17},
   Publisher = {Elsevier BV},
   Year = {1991},
   Month = {January},
   ISSN = {0304-4149},
   url = {http://dx.doi.org/10.1016/0304-4149(91)90056-I},
   Abstract = {We consider Markov processes ηt ⊂ Zd in which (i)
             particles die at rate δ ≥ 0, (ii) births from x to a
             neighboring y occur at rate 1, and (iii) when a new particle
             lands on an occupied site the particles annihilate each
             other and a vacant site results. When δ = 0 product measure
             with density 1 2 is a stationary distribution; we show it is
             the limit whenever P(η0≠ ø) = 1. We also show that if δ
             is small there is a nontrivial stationary distribution, and
             that for any δ there are most two extremal translation
             invariant stationary distributions. © 1991.},
   Doi = {10.1016/0304-4149(91)90056-I},
   Key = {fds243458}
}

@article{fds243453,
   Author = {Jinwen, C and Durrett, R and Xiufang, L},
   Title = {Exponential convergence for one dimensional contact
             processes},
   Journal = {Acta Mathematica Sinica, English Series},
   Volume = {6},
   Number = {4},
   Pages = {349-353},
   Publisher = {Springer Nature},
   Year = {1990},
   Month = {December},
   ISSN = {1439-8516},
   url = {http://dx.doi.org/10.1007/BF02107968},
   Abstract = {The complete convergence theorem implies that starting from
             any initial distribution the one dimensional contact process
             converges to a limit as t→∞. In this paper we give a
             necessary and sufficient condition on the initial
             distribution for the convergence to occur with exponential
             rapidity. © 1990 Springer-Verlag.},
   Doi = {10.1007/BF02107968},
   Key = {fds243453}
}

@article{fds243452,
   Author = {Ding, W-D and Durrett, R and Liggett, TM},
   Title = {Ergodicity of reversible reaction diffusion
             processes},
   Journal = {Probability Theory and Related Fields},
   Volume = {85},
   Number = {1},
   Pages = {13-26},
   Publisher = {Springer Nature},
   Year = {1990},
   Month = {March},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/bf01377624},
   Abstract = {Reaction-diffusion processes were introduced by Nicolis and
             Prigogine, and Haken. Existence theorems have been
             established for most models, but not much is known about
             ergodic properties. In this paper we study a class of models
             which have a reversible measure. We show that the stationary
             distribution is unique and is the limit starting from any
             initial distribution. © 1990 Springer-Verlag.},
   Doi = {10.1007/bf01377624},
   Key = {fds243452}
}

@article{fds243454,
   Author = {Cox, JT and Durrett, R},
   Title = {Large deviations for independent random walks},
   Journal = {Probability Theory and Related Fields},
   Volume = {84},
   Number = {1},
   Pages = {67-82},
   Publisher = {Springer Nature},
   Year = {1990},
   Month = {March},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/bf01288559},
   Abstract = {We consider a system of independent random walks on ℤ. Let
             ξn(x) be the number of particles at x at time n, and let
             Ln(x)=ξ0(x)+ ... +ξn(x) be the total occupation time of x
             by time n. In this paper we study the large deviations of
             Ln(0)-Ln(1). The behavior we find is much different from
             that of Ln(0). We investigate the limiting behavior when the
             initial configurations has asymptotic density 1 and when
             ξ0(x) are i.i.d Poisson mean 1, finding that the
             asymptotics are different in these two cases. © 1990
             Springer-Verlag.},
   Doi = {10.1007/bf01288559},
   Key = {fds243454}
}

@article{fds243450,
   Author = {Durrett, R and Tanaka, NI},
   Title = {Scaling inequalities for oriented percolation},
   Journal = {Journal of Statistical Physics},
   Volume = {55},
   Number = {5-6},
   Pages = {981-995},
   Publisher = {Springer Nature},
   Year = {1989},
   Month = {June},
   ISSN = {0022-4715},
   url = {http://dx.doi.org/10.1007/bf01041075},
   Abstract = {We look at seven critical exponents associated with
             two-dimensional oriented percolation. Scaling theory implies
             that these quantities satisfy four equalities. We prove five
             related inequalitites. © 1989 Plenum Publishing
             Corporation.},
   Doi = {10.1007/bf01041075},
   Key = {fds243450}
}

@article{fds243451,
   Author = {Durrett, R and Schonmann, RH and Tanaka, NI},
   Title = {Correlation lengths for oriented percolation},
   Journal = {Journal of Statistical Physics},
   Volume = {55},
   Number = {5-6},
   Pages = {965-979},
   Publisher = {Springer Nature},
   Year = {1989},
   Month = {June},
   ISSN = {0022-4715},
   url = {http://dx.doi.org/10.1007/bf01041074},
   Abstract = {Oriented percolation has two correlation lengths, one in the
             "space" and one in the "time" direction. In this paper we
             define these quantities for the two-dimensional model in
             terms of the exponential decay of suitably chosen
             quantities, and study the relationship between the various
             definitions. The definitions are used in a companion paper
             to prove inequalities between critical exponents. © 1989
             Plenum Publishing Corporation.},
   Doi = {10.1007/bf01041074},
   Key = {fds243451}
}

@article{fds243447,
   Author = {Bramson, M and Durrett, R},
   Title = {A simple proof of the stability criterion of Gray and
             Griffeath},
   Journal = {Probability Theory and Related Fields},
   Volume = {80},
   Number = {2},
   Pages = {293-298},
   Publisher = {Springer Nature America, Inc},
   Year = {1988},
   Month = {December},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/BF00356107},
   Abstract = {Gray and Griffeath studied attractive nearest neighbor spin
             systems on the integers having "all 0's" and "all 1's" as
             traps. Using the contour method, they established a
             necessary and sufficient condition for the stability of the
             "all 1's" equilibrium under small perturbations. In this
             paper we use a renormalized site construction to give a much
             simpler proof. Our new approach can be used in many
             situations as a substitute for the contour method. © 1988
             Springer-Verlag.},
   Doi = {10.1007/BF00356107},
   Key = {fds243447}
}

@article{fds243448,
   Author = {Cox, JT and Durrett, R},
   Title = {Limit theorems for the spread of epidemics and forest
             fires},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {30},
   Number = {2},
   Pages = {171-191},
   Publisher = {Elsevier BV},
   Year = {1988},
   Month = {December},
   ISSN = {0304-4149},
   url = {http://dx.doi.org/10.1016/0304-4149(88)90083-x},
   Abstract = {We prove that the "spatial epidemic with removal" grows
             linearly and has an asymptotic shape on the set of
             nonextinction. © 1988.},
   Doi = {10.1016/0304-4149(88)90083-x},
   Key = {fds243448}
}

@article{fds243446,
   Author = {Bramson, M and Durrett, R},
   Title = {Random walk in random environment: A counterexample?},
   Journal = {Communications in Mathematical Physics},
   Volume = {119},
   Number = {2},
   Pages = {199-211},
   Publisher = {Springer Nature},
   Year = {1988},
   Month = {June},
   ISSN = {0010-3616},
   url = {http://dx.doi.org/10.1007/bf01217738},
   Abstract = {We describe a family of random walks in random environments
             which have exponentially decaying correlations, nearest
             neighbor transition probabilities which are bounded away
             from 0, and yet are subdiffusive in any dimension d&lt;∞.
             © 1988 Springer-Verlag.},
   Doi = {10.1007/bf01217738},
   Key = {fds243446}
}

@article{fds323658,
   Author = {Durrett, R},
   Title = {Crabgrass, measles and gypsy moths: An introduction to
             modern probability},
   Journal = {Bulletin of the American Mathematical Society},
   Volume = {18},
   Number = {2},
   Pages = {117-144},
   Publisher = {American Mathematical Society (AMS)},
   Year = {1988},
   Month = {April},
   url = {http://dx.doi.org/10.1090/s0273-0979-1988-15625-x},
   Doi = {10.1090/s0273-0979-1988-15625-x},
   Key = {fds323658}
}

@article{fds243444,
   Author = {Durrett, R},
   Title = {Crabgrass, measles, and gypsy moths: An introduction to
             interacting particle systems},
   Journal = {The Mathematical Intelligencer},
   Volume = {10},
   Number = {2},
   Pages = {37-47},
   Publisher = {Springer Nature},
   Year = {1988},
   Month = {March},
   ISSN = {0343-6993},
   url = {http://dx.doi.org/10.1007/bf03028355},
   Doi = {10.1007/bf03028355},
   Key = {fds243444}
}

@article{fds243445,
   Author = {Chayes, JT and Chayes, L and Durrett, R},
   Title = {Connectivity properties of Mandelbrot's percolation
             process},
   Journal = {Probability Theory and Related Fields},
   Volume = {77},
   Number = {3},
   Pages = {307-324},
   Publisher = {Springer Nature America, Inc},
   Year = {1988},
   Month = {March},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/BF00319291},
   Abstract = {In 1974, Mandelbrot introduced a process in [0, 1]2 which he
             called "canonical curdling" and later used in this book(s)
             on fractals to generate self-similar random sets with
             Hausdorff dimension D∈(0,2). In this paper we will study
             the connectivity or "percolation" properties of these sets,
             proving all of the claims he made in Sect. 23 of the
             "Fractal Geometry of Nature" and a new one that he did not
             anticipate: There is a probability pc∈(0,1) so that if
             p<pc then the set is "duslike" i.e., the largest connected
             component is a point, whereas if p≧pc (notice the =)
             opposing sides are connected with positive probability and
             furthermore if we tile the plane with independent copies of
             the system then there is with probability one a unique
             unbounded connected component which intersects a positive
             fraction of the tiles. More succinctly put the system has a
             first order phase transition. © 1988 Springer-Verlag.},
   Doi = {10.1007/BF00319291},
   Key = {fds243445}
}

@article{fds243449,
   Author = {Durrett, R and Schonmann, RH},
   Title = {Large deviations for the contact process and two dimensional
             percolation},
   Journal = {Probability Theory and Related Fields},
   Volume = {77},
   Number = {4},
   Pages = {583-603},
   Publisher = {Springer Nature},
   Year = {1988},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/bf00959619},
   Abstract = {The following results are proved: 1) For the upper invariant
             measure of the basic one-dimensional supercritical contact
             process the density of 1's has the usual large deviation
             behavior: the probability of a large deviation decays
             exponentially with the number of sites considered. 2) For
             supercritical two-dimensional nearest neighbor site (or
             bond) percolation the density YΛ of sites inside a square
             Λ which belong to the infinite cluster has the following
             large deviation properties. The probability that YΛ
             deviates from its expected value by a positive amount decays
             exponentially with the area of Λ, while the probability
             that it deviates from its expected value by a negative
             amount decays exponentially with the perimeter of Λ. These
             two problems are treated together in this paper because
             similar techniques (renormalization) are used for both. ©
             1988 Springer-Verlag.},
   Doi = {10.1007/bf00959619},
   Key = {fds243449}
}

@article{fds243442,
   Author = {Brennan, MD and Durrett, R},
   Title = {Splitting intervals II: Limit laws for lengths},
   Journal = {Probability Theory and Related Fields},
   Volume = {75},
   Number = {1},
   Pages = {109-127},
   Publisher = {Springer Nature America, Inc},
   Year = {1987},
   Month = {May},
   ISSN = {0178-8051},
   url = {http://dx.doi.org/10.1007/BF00320085},
   Abstract = {In the processes under consideration, a particle of size L
             splits with exponential rate L α , 0<α<∞, and when it
             splits, it splits into two particles of size LV and L(1-V)
             where V is independent of the past with d.f. F on (0, 1).
             Let Z t be the number of particles at time t and L t the
             size of a randomly chosen particle. If α=0, it is well
             known how the system evolves: e -t Z t converges a.s. to an
             exponential r.v. and -L t ≈t + Ct 1/2 X where X is a
             standard normal t.v. Our results for α>0 are in sharp
             contrast. In "Splitting Intervals" we showed that t -1/α Z
             t converges a.s. to a constant K>0, and in this paper we
             show {Mathematical expression}. We show that the empirical
             d.f. of the rescaled lengths, {Mathematical expression},
             converges a.s. to a non-degenerate limit depending on F that
             we explicitly describe. Our results with α=2/3 are relevant
             to polymer degradation. © 1987 Springer-Verlag.},
   Doi = {10.1007/BF00320085},
   Key = {fds243442}
}

@article{fds243443,
   Author = {Chayes, JT and Chayes, L and Durrett, R},
   Title = {Inhomogeneous percolation problems and incipient infinite
             clusters},
   Journal = {Journal of Physics A: Mathematical and General},
   Volume = {20},
   Number = {6},
   Pages = {1521-1530},
   Publisher = {IOP Publishing},
   Year = {1987},
   Month = {April},
   ISSN = {0305-4470},
   url = {http://dx.doi.org/10.1088/0305-4470/20/6/034},
   Abstract = {The authors consider inhomogeneous percolation models with
             density p c+f(x) and examine the forms of f(x) which produce
             incipient structures. Taking f(x) approximately= mod x mod -
             lambda and assuming the existence of a correlation length
             exponent v for the homogeneous percolation model, they prove
             that in d=2, the borderline value of lambda is lambda b=1/v.
             If lambda &gt;1/v then, with probability one, there is no
             infinite cluster, while if lambda &lt;1/v then, with
             positive probability, the origin is part of an infinite
             cluster. This result sheds some light on numerical and
             theoretical predictions of certain properties of incipient
             infinite clusters. Furthermore, for d&gt;2, the models
             studied suggest what sort of 'incipient objects' should be
             examined in random surface models.},
   Doi = {10.1088/0305-4470/20/6/034},
   Key = {fds243443}
}

@article{fds243440,
   Author = {Chayes, JT and Chayes, L and Durrett, R},
   Title = {Critical behavior of the two-dimensional first passage
             time},
   Journal = {Journal of Statistical Physics},
   Volume = {45},
   Number = {5-6},
   Pages = {933-951},
   Publisher = {Springer Nature},
   Year = {1986},
   Month = {December},
   ISSN = {0022-4715},
   url = {http://dx.doi.org/10.1007/bf01020583},
   Abstract = {We study the two-dimensional first passage problem in which
             bonds have zero and unit passage times with probability p
             and 1-p, respectively. We prove that as the zero-time bonds
             approach the percolation threshold pc, the first passage
             time exhibits the same critical behavior as the correlation
             function of the underlying percolation problem. In
             particular, if the correlation length obeys ξ(p)
             ∼|p-pc|-v, then the first passage time constant satisfies
             μ(p)∼|p-pc|v. At pc, where it has been asserted that the
             first passage time from 0 to x scales as |x| to a power ψ
             with 0&lt;ψ&lt;1, we show that the passage times grow like
             log |x|, i.e., the fluid spreads exponentially rapidly. ©
             1986 Plenum Publishing Corporation.},
   Doi = {10.1007/bf01020583},
   Key = {fds243440}
}

@article{fds243441,
   Author = {Durrett, R},
   Title = {Multidimensional random walks in random environments with
             subclassical limiting behavior},
   Journal = {Communications in Mathematical Physics},
   Volume = {104},
   Number = {1},
   Pages = {87-102},
   Publisher = {Springer Nature},
   Year = {1986},
   Month = {March},
   ISSN = {0010-3616},
   url = {http://dx.doi.org/10.1007/bf01210794},
   Abstract = {In this paper we will describe and analyze a class of
             multidimensional random walks in random environments which
             contain the one dimensional nearest neighbor situation as a
             special case and have the pleasant feature that quite a lot
             can be said about them. Our results make rigorous a
             heuristic argument of Marinari et al. (1983), and show that
             in any d&lt;∞ we can have (a)Xn is recurrent and
             (b)Xn∼(log n)2. © 1986 Springer-Verlag.},
   Doi = {10.1007/bf01210794},
   Key = {fds243441}
}

@article{fds243439,
   Author = {Durrett, R},
   Title = {Some general results concerning the critical exponents of
             percolation processes},
   Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
             Gebiete},
   Volume = {69},
   Number = {3},
   Pages = {421-437},
   Publisher = {Springer Nature America, Inc},
   Year = {1985},
   Month = {September},
   ISSN = {0044-3719},
   url = {http://dx.doi.org/10.1007/BF00532742},
   Abstract = {In this paper we will give some results concerning the
             critical exponents of percolation processes which are valid
             for "any" model. These results show that in several respects
             the behavior which occurs for percolation on the binary tree
             provides bounds on one side for what happens in general.
             These results and their proofs are closely related to their
             analogues for the Ising model. © 1985 Springer-Verlag.},
   Doi = {10.1007/BF00532742},
   Key = {fds243439}
}

@article{fds243438,
   Author = {Durrett, R and Nguyen, B},
   Title = {Thermodynamic inequalities for percolation},
   Journal = {Communications in Mathematical Physics},
   Volume = {99},
   Number = {2},
   Pages = {253-269},
   Publisher = {Springer Nature},
   Year = {1985},
   Month = {June},
   ISSN = {0010-3616},
   url = {http://dx.doi.org/10.1007/bf01212282},
   Abstract = {In this paper we describe the percolation analogues of the
             Gibbs and Helmholtz potentials and use these quantities to
             prove some general inequalities concerning the critical
             exponents of percolation processes. © 1985
             Springer-Verlag.},
   Doi = {10.1007/bf01212282},
   Key = {fds243438}
}

@article{fds243437,
   Author = {Durrett, R and Liggett, TM},
   Title = {Fixed points of the smoothing transformation},
   Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
             Gebiete},
   Volume = {64},
   Number = {3},
   Pages = {275-301},
   Publisher = {Springer Nature},
   Year = {1983},
   Month = {September},
   ISSN = {0044-3719},
   url = {http://dx.doi.org/10.1007/BF00532962},
   Abstract = {Let W 1,..., W N be N nonnegative random variables and let
             {Mathematical expression} be the class of all probability
             measures on [0, ∞). Define a transformation T on
             {Mathematical expression} by letting Tμ be the distribution
             of W 1X1+ ... + W N X N, where the X i are independent
             random variables with distribution μ, which are independent
             of W 1,..., W N as well. In earlier work, first Kahane and
             Peyriere, and then Holley and Liggett, obtained necessary
             and sufficient conditions for T to have a nontrivial fixed
             point of finite mean in the special cases that the W i are
             independent and identically distributed, or are fixed
             multiples of one random variable. In this paper we study the
             transformation in general. Assuming only that for some γ>1,
             EW i γ <∞ for all i, we determine exactly when T has a
             nontrivial fixed point (of finite or infinite mean). When it
             does, we find all fixed points and prove a convergence
             result. In particular, it turns out that in the previously
             considered cases, T always has a nontrivial fixed point. Our
             results were motivated by a number of open problems in
             infinite particle systems. The basic question is: in those
             cases in which an infinite particle system has no invariant
             measures of finite mean, does it have invariant measures of
             infinite mean? Our results suggest possible answers to this
             question for the generalized potlatch and smoothing
             processes studied by Holley and Liggett. © 1983
             Springer-Verlag.},
   Doi = {10.1007/BF00532962},
   Key = {fds243437}
}

@article{fds243435,
   Author = {Durrett, R},
   Title = {Maxima of branching random walks},
   Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
             Gebiete},
   Volume = {62},
   Number = {2},
   Pages = {165-170},
   Publisher = {Springer Nature America, Inc},
   Year = {1983},
   Month = {June},
   ISSN = {0044-3719},
   url = {http://dx.doi.org/10.1007/BF00538794},
   Abstract = {In recent years several authors have obtained limit theorems
             for Ln, the location of the rightmost particle in a
             supercritical branching random walk but all of these results
             have been proved under the assumption that the offspring
             distribution has φ{symbol}(θ) = ∝ exp (θx)dF(x)<∞ for
             some θ>0. In this paper we investigate what happens when
             there is a slowly varying function K so that
             1-F(x)∼x}-qK(x) as x → ∞ and log (-x)F(x)→0 as
             x→-∞. In this case we find that there is a sequence of
             constants an, which grow exponentially, so that Ln/an
             converges weakly to a nondegenerate distribution. This
             result is in sharp contrast to the linear growth of Ln
             observed in the case φ{symbol}(θ)<∞. © 1983
             Springer-Verlag.},
   Doi = {10.1007/BF00538794},
   Key = {fds243435}
}

@article{fds243436,
   Author = {Chung, KL and Durrett, R and Zhao, Z},
   Title = {Extension of domains with finite gauge},
   Journal = {Mathematische Annalen},
   Volume = {264},
   Number = {1},
   Pages = {73-79},
   Publisher = {Springer Nature},
   Year = {1983},
   Month = {March},
   ISSN = {0025-5831},
   url = {http://dx.doi.org/10.1007/bf01458051},
   Doi = {10.1007/bf01458051},
   Key = {fds243436}
}

@article{fds323659,
   Author = {Cox, JT and Durrett, R},
   Title = {Oriented percolation in dimensions d ≥ 4: bounds and
             asymptotic formulas},
   Journal = {Mathematical Proceedings of the Cambridge Philosophical
             Society},
   Volume = {93},
   Number = {01},
   Pages = {151-151},
   Publisher = {Cambridge University Press (CUP)},
   Year = {1983},
   Month = {January},
   url = {http://dx.doi.org/10.1017/s0305004100060436},
   Doi = {10.1017/s0305004100060436},
   Key = {fds323659}
}

@article{fds243433,
   Author = {Durrett, R and Griffeath, D},
   Title = {Contact processes in several dimensions},
   Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
             Gebiete},
   Volume = {59},
   Number = {4},
   Pages = {535-552},
   Publisher = {Springer Nature America, Inc},
   Year = {1982},
   Month = {December},
   ISSN = {0044-3719},
   url = {http://dx.doi.org/10.1007/BF00532808},
   Doi = {10.1007/BF00532808},
   Key = {fds243433}
}

@article{fds243434,
   Author = {Durrett, R},
   Title = {An introduction to infinite particle systems},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {11},
   Number = {2},
   Pages = {109-150},
   Publisher = {Elsevier BV},
   Year = {1981},
   Month = {May},
   ISSN = {0304-4149},
   url = {http://dx.doi.org/10.1016/0304-4149(81)90001-6},
   Abstract = {In 1970, Spitzer wrote a paper called "Interaction of Markov
             processes" in which he introduced several classes of
             interacting particle systems. These processes and other
             related models, collectively referred to as infinite
             particle systems, have been the object of much research in
             the last ten years. In this paper we will survey some of the
             results which have been obtained and some of the open
             problems, concentrating on six overlapping classes of
             processes: the voter model, additive processes, the
             exponential family, one dimensional systems, attractive
             systems, and the Ising model. © 1981.},
   Doi = {10.1016/0304-4149(81)90001-6},
   Key = {fds243434}
}

@article{fds243432,
   Author = {Durrett, R},
   Title = {Conditioned limit theorems for random walks with negative
             drift},
   Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
             Gebiete},
   Volume = {52},
   Number = {3},
   Pages = {277-287},
   Publisher = {Springer Nature},
   Year = {1980},
   ISSN = {0044-3719},
   url = {http://dx.doi.org/10.1007/bf00538892},
   Abstract = {In this paper we will solve a problem posed by Iglehart. In
             (1975) he conjectured that if Sn is a random walk with
             negative mean and finite variance then there is a constant
             α so that (S[n.]/αn1/2|N&gt;n) converges weakly to a
             process which he called the Brownian excursion. It will be
             shown that his conjecture is false or, more precisely, that
             if ES1=-a&lt;0, ES12&lt;∞, and there is a slowly varying
             function L so that P(S1&gt;x)∼x-q L(x) as x→∞ then
             (S[n.]/n|Sn&gt;0) and (S[n.]/n|N&gt;n) converge weakly to
             nondegenerate limits. The limit processes have sample paths
             which have a single jump (with d.f. (1-(x/a)-q)+) and are
             otherwise linear with slope -a. The jump occurs at a
             uniformly distributed time in the first case and at t=0 in
             the second. © 1980 Springer-Verlag.},
   Doi = {10.1007/bf00538892},
   Key = {fds243432}
}

@article{fds243430,
   Author = {Durrett, R},
   Title = {Maxima of branching random walks vs. independent random
             walks},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {9},
   Number = {2},
   Pages = {117-135},
   Publisher = {Elsevier BV},
   Year = {1979},
   Month = {November},
   ISSN = {0304-4149},
   url = {http://dx.doi.org/10.1016/0304-4149(79)90024-3},
   Abstract = {In recent years several authors have obtained limit theorems
             for the location of the right most particle in a
             supercritical branching random walk. In this paper we will
             consider analogous problems for an exponentially growing
             number of independent random walks. A comparison of our
             results with the known results of branching random walk then
             identifies the limit behaviors which are due to the number
             of particles and those which are determined by the branching
             structure. © 1979.},
   Doi = {10.1016/0304-4149(79)90024-3},
   Key = {fds243430}
}

@article{fds243431,
   Author = {Anthony, PF and Durrett, R and Pulec, JL and Hartstone,
             JL},
   Title = {A new parameter in brain stem evoked response: Component
             wave areas},
   Journal = {Laryngoscope},
   Volume = {89},
   Number = {10 I},
   Pages = {1569-1578},
   Year = {1979},
   Abstract = {Using a newly developed brain stem evoked response (BSER)
             parameter in preliminary testing, the authors can
             individually identify the auditory thresholds of 500 Hz,
             1000 Hz and 2000 Hz to within 15-25 db. The authors have
             developed an accurate method of breaking down the complex
             BSER curve, using a successive approximation technique, into
             its individual component curves. Each component curve is
             thought to represent the isolated electrical activity of one
             generator site. The component curves are the shape of normal
             distribution curves. The area under each individual
             component curve is the new parameter which the authors feel
             represents the isolated electrical activity of one generator
             site. Using this parameter a clinical trial was performed.
             Constant ipsilateral pure tone masking was superimposed upon
             the stimulus clicks in the test ears of two subjects. The
             constant ipsilateral masking was superimposed at 500 Hz,
             1000 Hz and 2000 Hz. A statistically significant decrease in
             the area of component wave II (one masking sound caused an
             increased area) was seen when the pure tone masking sound
             became at least 15-25 db louder than the patient's threshold
             at that individual frequency. These preliminary results give
             reason to think that a method of quantification of BSER
             responses has been found. More importantly, a method of
             identifying individual audiometric thresholds from 500 Hz
             throughout 2000 Hz to within 15-25 db has been found. These
             findings need extensive further testing.},
   Key = {fds243431}
}

@article{fds243428,
   Author = {Durrett, R},
   Title = {The genealogy of critical branching processes},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {8},
   Number = {1},
   Pages = {101-116},
   Publisher = {Elsevier BV},
   Year = {1978},
   Month = {November},
   ISSN = {0304-4149},
   url = {http://dx.doi.org/10.1016/0304-4149(78)90071-6},
   Abstract = {In this paper we will obtain results concerning the
             distribution of generations and the degree of relationship
             of the individuals in a critical branching process {Z(t),
             t≥0} and we will apply these results to obtain a "central
             limit theorem" for critical branching random walks. ©
             1978.},
   Doi = {10.1016/0304-4149(78)90071-6},
   Key = {fds243428}
}

@article{fds243429,
   Author = {Durrett, RT and Resnick, SI},
   Title = {Weak convergence with random indices},
   Journal = {Stochastic Processes and Their Applications},
   Volume = {5},
   Number = {3},
   Pages = {213-220},
   Publisher = {Elsevier BV},
   Year = {1977},
   Month = {January},
   ISSN = {0304-4149},
   url = {http://dx.doi.org/10.1016/0304-4149(77)90031-X},
   Abstract = {Suppose {Xnn≥-0} are random variables such that for
             normalizing constants an>0, bn, n≥0 we have Yn(·)=(X[n,
             ·]-bn/an ⇒ Y(·) in D(0.∞) . Then an and bn must in
             specific ways and the process Y possesses a scaling
             property. If {Nn} are positive integer valued random
             variables we discuss when YNn → Y and Y'n=(X[Nn]-bn)/an
             ⇒ Y'. Results given subsume random index limit theorems
             for convergence to Brownian motion, stable processes and
             extremal processes. © 1977.},
   Doi = {10.1016/0304-4149(77)90031-X},
   Key = {fds243429}
}

@article{fds243426,
   Author = {Chung, KL and Durrett, R},
   Title = {Downcrossings and local time},
   Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
             Gebiete},
   Volume = {35},
   Number = {2},
   Pages = {147-149},
   Publisher = {Springer Nature America, Inc},
   Year = {1976},
   Month = {June},
   ISSN = {0044-3719},
   url = {http://dx.doi.org/10.1007/BF00533319},
   Doi = {10.1007/BF00533319},
   Key = {fds243426}
}

@article{fds243427,
   Author = {Durrett, RT and Ghurye, SG},
   Title = {WAITING TIMES WITHOUT MEMORY.},
   Journal = {J Appl Probab},
   Volume = {13},
   Number = {1},
   Pages = {65-75},
   Year = {1976},
   Abstract = {A waiting time without memory, or age-independent residual
             life-time, is a positive-valued random variable T with the
             property that for any x, y greater than 0, given that T
             greater than x, the conditional probability of T greater
             than x plus y is the same as the unconditional probability
             of T greater than y; in other words, the physical process
             operates as if it has no memory concerning the successive
             occurrences of a certain event. The paper investigates the
             consequences of defining the property of lack of memory on
             more general time-domains than the positive reals. As a side
             issue, there is discussion of a stochastic variation of
             Cauchy's functional equation.},
   Key = {fds243427}
}


%% Papers Accepted   
@article{fds220503,
   Author = {R. Durrett and S. Moseley},
   Title = {Spatial Moran Models. I. Tunneling in the Neutral
             Case},
   Journal = {Annals Applied Probability},
   Year = {2013},
   Key = {fds220503}
}


%% Papers Submitted   
@article{fds220504,
   Author = {S. Magura and V. Pong and D. Sivakoff and R. Durrett},
   Title = {Two evolving social network models},
   Year = {2013},
   Key = {fds220504}
}

@article{fds211319,
   Author = {R. Durrett},
   Title = {Phase transition in a meta-population version of Schelling's
             model},
   Year = {2012},
   Key = {fds211319}
}

@article{fds211320,
   Author = {R. Durrett and J. Foo and K. Leder},
   Title = {Spatial Moran Models II. Tumor growth and
             progression},
   Year = {2012},
   Key = {fds211320}
}

 

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ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320