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 $\mathbb{Z}^d$, $d \geq 1$, and $\xi$ is a simple symmetric exclusion process on $\mathbb{Z}^d$ in Bernoulli equilibrium. This model describes the evolution of a reactant $u$ under the influence of a catalyst $\xi$.

In Gärtner, den Hollander and Maillard [3] 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 critical dimension $d=3$, which was left open in Gärtner, den Hollander and Maillard [3] and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a Green term, as in $d\geq 4$, but also by a polaron term. The presence of the latter implies intermittency of all orders above a finite threshold for $\kappa$.