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.

翻译：我们考虑一个由$n$个粒子组成的系统，这些粒子在实轴上以跳跃方式向前运动。系统状态由粒子位置的经验分布描述。每个粒子在某些时间点“向前跳跃”，其瞬时跳跃率由粒子在当前状态（经验分布）中的位置分位数的递减函数给出。已有研究在特定条件下建立了当$n\to\infty$时系统随机动力学收敛到确定性平均场模型（MFM）的结果，该模型是积分-微分方程的解。另一系列已有工作证明了行波形式的MFM的存在性，以及MFM轨迹向行波的吸引性。本文的主要结果包括：（a）我们证明当$n\to\infty$时，（重中心化）状态的平稳分布集中在一个（重中心化）行波上；（b）我们得到了关于（重中心化）状态平稳分布的一个关于$n$一致的矩界；（c）我们证明了一个收敛到MFM的结果，该结果相比于已有工作具有显著的一般性。结果（b）和（c）既是证明（a）的“组成部分”，也独立具有研究价值。