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

We consider the (discrete) parabolic Anderson model ∂u(t,x)∕∂t=Δu(t,x)+ ξt(x)u(t,x), t≥0, x∈Zd. Here, the ξ-field is R-valued, acting as a dynamic random environment, and Δ represents the discrete Laplacian. We focus on the case where ξ 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 [5], 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.