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

Dirk Erhard, Martin Hairer, Tiecheng Xu

Dirk Erhard, Martin Hairer, Tiecheng Xu

**26/3/24**Published in : arXiv:2403.17669

We consider the (discrete) parabolic Anderson model \partial u(t,x)/\partial t=\Delta u(t,x) +\xi_t(x) u(t,x), t≥0, x\in \mathbb{Z}^d. Here, the \xi-field is R-valued, acting as a dynamic random environment, and Δ 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.