Intermittency on catalysts: three-dimensional simple symmetric exclusion

Abstract

We continue our study of intermittency for the parabolic Anderson model $\partial u/\partial t=\kappa \Delta u+\xi u$ in a space-time random medium $\xi$, where $\kappa$ is a positive diffusion constant, $\Delta$ is the lattice Laplacian on ${\mathbb{Z}}^d$, $d\ge 1$, and $\xi$ is a simple symmetric exclusion process on ${\mathbb{Z}}^d$ in Bernoulli equilibrium. This model describes the evolution of a \emph{reactant} $u$ under the influence of a \emph{catalyst} $\xi$. In Gärtner, den Hollander and Maillard (2007) we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as $t\to \infty$ of the successive moments of the solution $u$. This led to an almost complete picture of intermittency as a function of $d$ and $\kappa$. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as $\kappa \to \infty$ in the \emph{critical} dimension $d=3$, which was left open in Gärtner, den Hollander and Maillard (2007) and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a \emph{Green} term, as in $d\ge 4$, but also by a \emph{polaron} term. The presence of the latter implies intermittency of \emph{all} orders above a finite threshold for $\kappa$.

38 pages