A scaling limit of the parabolic Anderson model with exclusion interaction

TL;DR

This paper studies the scaling limit of the parabolic Anderson model with exclusion interaction, showing convergence of solutions in three dimensions and deriving asymptotics for a related random walk survival probability.

Contribution

It introduces a novel analysis of the parabolic Anderson model with exclusion interaction using regularity structures and joint cumulant estimates, advancing understanding of its scaling limits.

Findings

Solutions converge in law in three dimensions with proper renormalization.

Derived precise asymptotics for the survival probability of a killed random walk.

Developed new sharp estimates for joint cumulants of the exclusion process.

Abstract

We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=\Delta u(t,x) +\xi_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$, 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.