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

A scaling limit of the parabolic Anderson model with exclusion interaction