Astérisque N° 349/2013 - Grand Format

We consider particle systems that are perturbations of the voter model and show that when space and time are resealed 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 super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial 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 Lotka-Volterra model of Neuhauser and Pacata, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, (iii) a continuous time version of the non-linear 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 ("Survival and coexistence in stochastic spatial Lotka-Volterra models", 2007) and the second confirms a conjecture of Ohtsuki et al. ("A simple rule for the evolution of cooperation on graphs and social networks", 2006) in the context of certain infinit graphs. An important feature of our general results is that do not require the process to be attractive.