Intermittency on catalysts: symmetric exclusion

Abstract

We continue our study of intermittency for the parabolic Anderson equation $\partial u/\partial t = \kappa\Delta u + \xi u$, where $u\colon \Z^d\times [0,\infty)\to\R$, $\kappa$ is the diffusion constant, $\Delta$ is the discrete Laplacian, and $\xi\colon \Z^d\times [0,\infty)\to\R$ is a space-time random medium. The solution of the equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $\xi$. In this paper we focus on the case where $\xi$ is exclusion with a symmetric random walk transition kernel, starting from equilibrium with density $\rho\in (0,1)$. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant $\kappa$ when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents for $\kappa\to\infty$, which is controlled by moderate deviations of $\xi$ requiring a delicate expansion argument. In G\"artner and den Hollander \cite{garhol04} the case where $\xi$ is a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role.