A particle system with mean-field interaction: Large-scale limit of stationary distributions

TL;DR

This paper studies a particle system with mean-field interactions, proving that as the number of particles grows, the stationary distributions concentrate on traveling waves, with uniform moment bounds and convergence to the mean-field model.

Contribution

It establishes the large-scale limit of stationary distributions concentrating on traveling waves and provides uniform moment bounds and generalized convergence results.

Findings

Stationary distributions concentrate on traveling waves as particle number increases.

Uniform moment bounds are established for stationary distributions.

Convergence of the system to the mean-field model is proven under broader conditions.

Abstract

We consider a system consisting of $n$ particles, moving forward in jumps on the real line. System state is the empirical distribution of particle locations. Each particle ``jumps forward'' at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle's location quantile within the current state (empirical distribution). Previous work on this model established, under certain conditions, the convergence, as $n\to\infty$, of the system random dynamics to that of a deterministic mean-field model (MFM), which is a solution to an integro-differential equation. Another line of previous work established the existence of MFMs that are traveling waves, as well as the attraction of MFM trajectories to traveling waves. The main results of this paper are: (a) We prove that, as $n\to\infty$, the stationary distributions of (re-centered) states concentrate on a (re-centered) traveling wave; (b) We obtain a uniform across $n$ moment bound on the stationary distributions of (re-centered) states; (c) We prove a convergence-to-MFM result, which is substantially more general than that in previous work. Results (b) and (c) serve as ``ingredients'' of the proof of (a), but also are of independent interest.