A scaling limit of the parabolic Anderson model with exclusion interaction

Abstract

We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=\Delta u(t,x)+{\xi }_t(x)u(t,x)$, $t\ge 0$, $x\in {\mathbb{Z}}^d$, where the $\xi$-field is $\mathbb{R}$-valued and plays the role of a dynamic random environment, and $\Delta$ is the discrete Laplacian. We focus on the case in which $\xi$ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein--Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension $d=3$ upon considering a suitably renormalised version of the above equation, the sequence of solutions converges in law. As a by-product of our main result we obtain precise asymptotics for the survival probability of a simple random walk that is killed at a scale dependent rate when meeting an exclusion particle. Our proof relies on the discrete theory of regularity structures of \cite{ErhardHairerRegularity} and on novel sharp estimates of joint cumulants of arbitrary large order for the exclusion process. We think that the latter is of independent interest and may find applications elsewhere.