Intermittency on catalysts: three-dimensional simple symmetric exclusion

We continue our study of intermittency for the parabolic Anderson model $\partial u/\partial t = κΔu + ξu$ in a space-time random medium $ξ$, where $κ$ is a positive diffusion constant, $Δ$ is the lattice Laplacian on $\Z^d$, $d \geq 1$, and $ξ$ 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} $ξ$. 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 $κ$. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as $κ\to\infty$ 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 $d\geq 4$, but also by a \emph{polaron} term. The presence of the latter implies intermittency of \emph{all} orders above a finite threshold for $κ$.