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 ℤd: 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→∞ the normalised empirical measure converges in D([0,T],ℳ1(ℝd)) 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.
Related
- Tamed Euler Approximation of Invariant Measures for Weakly Interacting Particle Systems and McKean--Vlasov Equations
- Feedback approaches for set-point stabilization of interacting particle systems
- Parameter estimation for graphon-interacting particle systems from discrete observations
- Path space large deviation on a lattice for interacting particle systems
- Drift parameter estimation for interacting particle systems with singular repulsion
BibTeX
@misc{ayala2026atomic,
title = {{Atomic limits of empirical measures of interacting particle systems with self-duality}},
author = {Mario Ayala and Nicolas Dirr and Johannes Zimmer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31319},
eprint = {2610.31319},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31319},
}
RIS
TY - UNPB
TI - Atomic limits of empirical measures of interacting particle systems with self-duality
AU - Mario Ayala
AU - Nicolas Dirr
AU - Johannes Zimmer
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31319
UR - https://arxiv.org/abs/2610.31319
AB - We study the fixed-particle limit of empirical measures for a unified self-dual family of conservative particle systems on 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→∞ the normalised empirical measure converges in D([0,T],ℳ_1(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.
ER -