Intermittency on catalysts: three-dimensional simple symmetric exclusion

We continue our study of intermittency for the parabolic Anderson model partial derivative u/partial derivative 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 >= 1, and xi is a simple symmetric exclusion process on Z(d) in Bernoulli equilibrium. This model describes the evolution of a reactant u under the influence of a catalyst xi. In [3] we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as t -> infinity 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 -> infinity in the critical dimension d = 3, which was left open in [3] and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a Green term, as in d >= 4, but also by a polaron term. The presence of the latter implies intermittency of all orders above a finite threshold for kappa.