%% Papers Published
@article{fds341500,
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 = {fds341500}
}
@article{fds344819,
Author = {Beckman, E and Frank, N and Jiang, Y and Junge, M and Tang,
S},
Title = {The frog model on trees with drift},
Journal = {Electronic Communications in Probability},
Volume = {24},
Year = {2019},
Month = {January},
url = {http://dx.doi.org/10.1214/19ECP235},
Abstract = {© 2019, Institute of Mathematical Statistics. All rights
reserved. We provide a uniform upper bound on the minimal
drift so that the onepersite frog model on a dary tree is
recurrent. To do this, we introduce a subprocess that
couples across trees with different degrees. Finding
couplings for frog models on nested sequences of graphs is
known to be difficult. The upper bound comes from combining
the coupling with a new, simpler proof that the frog model
on a binary tree is recurrent when the drift is sufficiently
strong. Additionally, we describe a coupling between frog
models on trees for which the degree of the smaller tree
divides that of the larger one. This implies that the
critical drift has a limit as d tends to infinity along
certain subsequences.},
Doi = {10.1214/19ECP235},
Key = {fds344819}
}
@article{fds344820,
Author = {Dygert, B and Kinzel, C and Junge, M and Raymond, A and Slivken, E and Zhu,
J},
Title = {The bullet problem with discrete speeds},
Journal = {Electronic Communications in Probability},
Volume = {24},
Year = {2019},
Month = {January},
url = {http://dx.doi.org/10.1214/19ECP238},
Abstract = {© 2019, Institute of Mathematical Statistics. All rights
reserved. Bullets are fired from the origin of the positive
real line, one per second, with independent speeds sampled
uniformly from a discrete set. Collisions result in mutual
annihilation. We show that a bullet with the second largest
speed survives with positive probability, while a bullet
with the smallest speed does not. This also holds for
exponential spacings between firing times. Our results imply
that the middlevelocity particle survives with positive
probability in a twosided version of the bullet process
with three speeds known to physicists as ballistic
annihilation.},
Doi = {10.1214/19ECP238},
Key = {fds344820}
}
@article{fds340352,
Author = {Brito, G and Fowler, C and Junge, M and Levy, A},
Title = {Ewens Sampling and Invariable Generation},
Journal = {Combinatorics, Probability and Computing},
Volume = {27},
Number = {6},
Pages = {853891},
Publisher = {Cambridge University Press (CUP)},
Year = {2018},
Month = {November},
url = {http://dx.doi.org/10.1017/S096354831800007X},
Abstract = {© 2018 Cambridge University Press. We study the number of
random permutations needed to invariably generate the
symmetric group Sn when the distribution of cycle counts has
the strong αlogarithmic property. The canonical example is
the Ewens sampling formula, for which the special case α =
1 corresponds to uniformly random permutations. For strong
αlogarithmic measures and almost every α, we show that
precisely [(1αlog2)1] permutations are needed to
invariably generate Sn with asymptotically positive
probability. A corollary is that for many other probability
measures on Sn no fixed number of permutations will
invariably generate Sn with positive probability. Along the
way we generalize classic theorems of ErdÅ's, Tehran,
Pyber, Łuczak and Bovey to permutations obtained from the
Ewens sampling formula.},
Doi = {10.1017/S096354831800007X},
Key = {fds340352}
}
@article{fds335538,
Author = {Johnson, T and Junge, M},
Title = {Stochastic orders and the frog model},
Journal = {Annales De L'Institut Henri Poincaré, Probabilités Et
Statistiques},
Volume = {54},
Number = {2},
Pages = {10131030},
Publisher = {Institute of Mathematical Statistics},
Year = {2018},
Month = {May},
url = {http://dx.doi.org/10.1214/17AIHP830},
Abstract = {© Association des Publications de l’Institut Henri
Poincaré, 2018. The frog model starts with one active
particle at the root of a graph and some number of dormant
particles at all nonroot vertices. Active particles follow
independent random paths, waking all inactive particles they
encounter. We prove that certain frog model statistics are
monotone in the initial configuration for two nonstandard
stochastic dominance relations: the increasing concave and
the probability generating function orders. This extends
many canonical theorems. We connect recurrence for random
initial configurations to recurrence for deterministic
configurations. Also, the limiting shape of activated sites
on the integer lattice respects both of these orders. Other
implications include monotonicity results on transience of
the frog model where the number of frogs per vertex decays
away from the origin, on survival of the frog model with
death, and on the time to visit a given vertex in any frog
model.},
Doi = {10.1214/17AIHP830},
Key = {fds335538}
}
@article{fds338420,
Author = {Foxall, E and Hutchcroft, T and Junge, M},
Title = {Coalescing random walk on unimodular graphs},
Journal = {Electronic Communications in Probability},
Volume = {23},
Publisher = {Institute of Mathematical Statistics},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1214/18ECP136},
Abstract = {© 2018, University of Washington. All rights reserved.
Coalescing random walk on a unimodular random rooted graph
for which the root has finite expected degree visits each
site infinitely often almost surely. A corollary is that an
opinion in the voter model on such graphs has infinite
expected lifetime. Additionally, we deduce an adaptation of
our main theorem that holds uniformly for coalescing random
walk on finite random unimodular graphs with degree
distribution stochastically dominated by a probability
measure with finite mean.},
Doi = {10.1214/18ECP136},
Key = {fds338420}
}
@article{fds339580,
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/10.1214/18EJP215},
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 = {10.1214/18EJP215},
Key = {fds339580}
}
@article{fds339742,
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},
Publisher = {Institute of Mathematical Statistics},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1214/18ECP182},
Abstract = {© 2018, University of Washington. All rights reserved. Fix
a probability distribution p = (p1, p2, …) on the positive
integers. The first block in a pbiased 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 = {10.1214/18ECP182},
Key = {fds339742}
}
@article{fds329100,
Author = {Hoffman, C and Johnson, T and Junge, M},
Title = {Recurrence and transience for the frog model on
trees},
Journal = {The Annals of Probability},
Volume = {45},
Number = {5},
Pages = {28262854},
Publisher = {Institute of Mathematical Statistics},
Year = {2017},
Month = {September},
url = {http://dx.doi.org/10.1214/16AOP1125},
Abstract = {© Institute of Mathematical Statistics, 2017. The frog
model is a growing system of random walks where a particle
is added whenever a new site is visited. A longstanding open
question is how often the root is visited on the infinite
dary tree. We prove the model undergoes a phase transition,
finding it recurrent for d = 2 and transient for d ≥ 5.
Simulations suggest strong recurrence for d = 2, weak
recurrence for d = 3, and transience for d ≥ 4.
Additionally, we prove a 01 law for all dary trees, and we
exhibit a graph on which a 01 law does not hold. To prove
recurrence when d = 2, we construct a recursive
distributional equation for the number of visits to the root
in a smaller process and show the unique solution must be
infinity a.s. The proof of transience when d = 5 relies on
computer calculations for the transition probabilities of a
large Markov chain. We also include the proof for d ≥ 6,
which uses similar techniques but does not require computer
assistance.},
Doi = {10.1214/16AOP1125},
Key = {fds329100}
}
@article{fds325463,
Author = {Hoffman, C and Johnson, T and Junge, M},
Title = {From transience to recurrence with poisson tree
frogs},
Journal = {The Annals of Applied Probability},
Volume = {26},
Number = {3},
Pages = {16201635},
Publisher = {Institute of Mathematical Statistics},
Year = {2016},
Month = {June},
url = {http://dx.doi.org/10.1214/15AAP1127},
Abstract = {© 2016 Institute of Mathematical Statistics. Consider the
following interacting particle system on the dary tree,
known as the frog model: Initially, one particle is awake at
the root and i.i.d. Poisson many particles are sleeping at
every other vertex. Particles that are awake perform simple
random walks, awakening any sleeping particles they
encounter. We prove that there is a phase transition between
transience and recurrence as the initial density of
particles increases, and we give the order of the transition
up to a logarithmic factor.},
Doi = {10.1214/15AAP1127},
Key = {fds325463}
}
@article{fds325464,
Author = {Benjamini, I and Foxall, E and GurelGurevich, O and Junge, M and Kesten, H},
Title = {Site recurrence for coalescing random walk},
Journal = {Electronic Communications in Probability},
Volume = {21},
Publisher = {Institute of Mathematical Statistics},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.1214/16ECP5},
Abstract = {© 2016, University of Washington. All rights reserved.
Begin continuous time random walks from every vertex of a
graph and have particles coalesce when they collide. We use
a duality relation with the voter model to prove the process
is site recurrent on bounded degree graphs, and for
GaltonWatson trees whose offspring distribution has
exponential tail. We prove bounds on the occupation
probability of a site, as well as a general 01 law. Similar
conclusions hold for a coalescing process on trees where
particles do not backtrack.},
Doi = {10.1214/16ECP5},
Key = {fds325464}
}
@article{fds325465,
Author = {Johnson, T and Junge, M},
Title = {The critical density for the frog model is the degree of the
tree},
Journal = {Electronic Communications in Probability},
Volume = {21},
Publisher = {Institute of Mathematical Statistics},
Year = {2016},
Month = {January},
url = {http://dx.doi.org/10.1214/16ECP29},
Abstract = {© 2016, University of Washington. All rights reserved. The
frog model on the rooted dary tree changes from transient
to recurrent as the number of frogs per site is increased.
We prove that the location of this transition is on the same
order as the degree of the tree.},
Doi = {10.1214/16ECP29},
Key = {fds325465}
}
@article{fds325466,
Author = {Junge, M},
Title = {Choices, intervals and equidistribution},
Journal = {Electronic Journal of Probability},
Volume = {20},
Pages = {118},
Publisher = {Institute of Mathematical Statistics},
Year = {2015},
Month = {September},
url = {http://dx.doi.org/10.1214/EJP.v204191},
Abstract = {© 2015, University of Washington. All right reserved. We
give a sufficient condition for a random sequence in [0,1]
generated by a Ψ process to be equidistributed. The
condition is met by the canonical example – the max2
process – where the nth term is whichever of two uniformly
placed points falls in the larger gap formed by the previous
n — 1 points. This solves an open problem from Itai
Benjamini, Pascal Maillard and Elliot Paquette. We also
deduce equidistribution for more general Ψprocesses. This
includes an interpolation of the min2 and max2 processes
that is biased towards min2.},
Doi = {10.1214/EJP.v204191},
Key = {fds325466}
}
