A Particle System with Mean-Field Interaction: Large-Scale Limit of Stationary Distributions

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