INTERMITTENCY ON CATALYSTS: VOTER MODEL

• 成果类型: Article
• 署名作者: Gaertner, J.; den Hollander, F.; Maillard, G.
• 署名单位: Technical University of Berlin; Leiden University; Leiden University - Excl LUMC; Aix-Marseille Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP535
• 发表日期: 2010
• 页码: 2066-2102
• 关键词: symmetric exclusion

摘要

In this paper we study intermittency for the parabolic Anderson equation partial derivative u/partial derivative t = kappa Delta u + gamma xi u with u:Z(d) x [0, infinity) -> R, where kappa is an element of [0, infinity) is the diffusion constant. Delta is the discrete Laplacian, gamma is an element of (0, infinity) is the coupling constant, and xi : Z(d) x [0, infinity) -> R is a space-time random medium. The solution of this equation describes the evolution of a reactant u under the influence of a catalyst xi. We focus on the case where xi is the voter model with opinions 0 and 1 that are updated according to a random walk transition kernel, starting from either the Bernoulli measure nu(rho) or the equilibrium measure mu(rho), where rho is an element of (0, 1) is the density of 1's. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of u. We show that if the random walk transition kernel has zero mean and finite variance, then these exponents are trivial for 1 <= d <= 4, but display an interesting dependence on the diffusion constant kappa for d >= 5, with qualitatively different behavior in different dimensions. In earlier work we considered the case where xi is a field of independent simple random walks in a Poisson equilibrium, respectively, a symmetric exclusion process in a Bernoulli equilibrium, which are both reversible dynamics. In the present work a main obstacle is the nonreversibility of the voter model dynamics, since this precludes the application of spectral techniques. The duality with coalescing random walks is key to our analysis, and leads to a representation formula for the Lyapunov exponents that allows for the application of large deviation estimates.