Math @ Duke

Publications [#243527] of Richard T. Durrett
Papers Published
 Cox, JT; Durrett, R; Perkins, EA, Voter model perturbations and reaction diffusion equations,
Astérisque, vol. 349 no. 349
(January, 2013),
pp. 1127 [math.PR/1103.1676]
(last updated on 2019/07/23)
Abstract: We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions d ≥ 3. Combining this result with properties of the P.D.E., some methods arising from a low density superBrownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of nontrivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of four systems when the parameters are close to the voter model: (i) a stochastic spatial LotkaVolterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, (iii) a continuous time version of the nonlinear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin, (iv) a voter model in which opinion changes are followed by an exponentially distributed latent period during which voters will not change again. The first application confirms a conjecture of Cox and Perkins [8] and the second confirms a conjecture of Ohtsuki et al. [34] in the context of certain infinite graphs. An important feature of our general results is that they do not require the process to be attractive.


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