A scaling limit of the 2D parabolic Anderson model with exclusion interaction

We consider the (discrete) parabolic Anderson model $\mathrm{\partial }u(t,x)\mathrm{/}\mathrm{\partial }t=\mathrm{\Delta }u(t,x)+{\xi }_t(x)u(t,x)$, $t\ge 0$, $x\in {\mathbb{Z}}^d$. Here, the $\xi$-field is $\mathbb{R}$-valued, acting as a dynamic random environment, and $\mathrm{\Delta }$ represents the discrete Laplacian. We focus on the case where $\xi$ is given by a rescaled symmetric simple exclusion process which converges to an Ornstein–Uhlenbeck process. By scaling the Laplacian diffusively and considering the equation on a torus, we demonstrate that in dimension $d=2$, when a suitably renormalized version of the above equation is considered, the sequence of solutions converges in law. This resolves an open problem from~\cite{EH23}, where a similar result was shown in the three-dimensional case. The novel contribution in the present work is the establishment of a gradient bound on the transition probability of a fixed but arbitrary number of labelled exclusion particles.