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

Alexander Stolyar
Jun 2022
摘要
We consider a system consisting of $n$ particles, moving forward in jumps onthe real line. System state is the empirical distribution of particlelocations. Each particle ``jumps forward'' at some time points, with theinstantaneous rate of jumps given by a decreasing function of the particle'slocation quantile within the current state (empirical distribution). Previouswork on this model established, under certain conditions, the convergence, as$n\to\infty$, of the system random dynamics to that of a deterministicmean-field model (MFM), which is a solution to an integro-differentialequation. Another line of previous work established the existence of MFMs thatare traveling waves, as well as the attraction of MFM trajectories to travelingwaves. 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 boundon the stationary distributions of (re-centered) states; (c) We prove aconvergence-to-MFM result, which is substantially more general than that inprevious work. Results (b) and (c) serve as ``ingredients'' of the proof of(a), but also are of independent interest.
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