Atomic limits of empirical measures of interacting particle systems with self-duality

Abstract

We study the fixed-particle limit of empirical measures for a unified self-dual family of conservative particle systems on $\mathbb{Z}^d$: independent random walkers, the symmetric exclusion process, and the symmetric inclusion process. For a fixed number of particles $k$, under diffusive space-time scaling parametrised by $n$, we prove that as $n\to\infty$ the normalised empirical measure converges in $D([0,T],\mathcal{M}_1(\mathbb{R}^d))$ to an atomic empirical measure carried by $k$ independent diffusions, with constant covariance determined by the microscopic jump kernel. Equivalently, the limit is the unique atomic solution of the Dean-Kawasaki martingale problem with mass quantisation $1/k$. In particular, exclusion and inclusion have the same fixed-$k$ atomic limit as independent walkers: the interaction contributes only through microscopic near-collisions, whose rescaled occupation time vanishes. The proof combines self-duality, consistency-based reduction to two particles, and Green-function estimates for finite perturbations of symmetric random walks.