Intermittency on catalysts: three-dimensional simple symmetric exclusion

TL;DR

This paper analyzes the intermittency behavior of a reactant evolving under a catalyst modeled by a symmetric exclusion process, focusing on the asymptotics of Lyapunov exponents in three dimensions as the diffusion constant grows large.

Contribution

It completes the study of Lyapunov exponents for the model in three dimensions, revealing the role of both Green and polaron terms in the asymptotics as diffusion increases.

Findings

Asymptotics in 3D involve Green and polaron terms.

Intermittency persists for all orders above a threshold.

Results extend understanding of reactant-catalyst dynamics in random media.

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 $\Z^d$, $d \geq 1$, and $\xi$ is a simple symmetric exclusion process on $\Z^d$ in Bernoulli equilibrium. This model describes the evolution of a \emph{reactant} $u$ under the influence of a \emph{catalyst} $\xi$. In G\"artner, 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 G\"artner, 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 $d\geq 4$, but also by a \emph{polaron} term. The presence of the latter implies intermittency of \emph{all} orders above a finite threshold for $\kappa$.