Intermittency on catalysts : symmetric exclusion

We continue our study of intermittency for the parabolic Anderson equation ∂u/∂t = κΔu + ξu, where u : Z × [0,∞) → R, κ is the diffusion constant, Δ is the discrete Laplacian, and ξ : Z × [0,∞) → 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” ξ. In this paper we focus on the case where ξ is exclusion with a symmetric random walk transition kernel, starting from equilibrium with density ρ ∈ (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 κ when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents for κ → ∞, which is controlled by moderate deviations of ξ requiring a delicate expansion argument. In Gärtner and den Hollander [4] the case where ξ 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. MSC 2000. Primary 60H25, 82C44; Secondary 60F10, 35B40.