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：系统状态是粒子点的经验分布。 每一个粒子“ 跳跃” 在某个时间点点上, 粒子位置微小的微小功能在目前状态( 经验分布 ) 中给的瞬时跳速率, 这个模型以前的工作在某些条件下, 以 $\ to\\ infty 美元 的形式建立, 系统随机动态的趋同, 与确定性平均地模型( MFM ) 的相近性( MFM ) 的相近性( MFM ) 是一个解决点。 之前的另外一线工作确定了移动波的MFM 的存在, 以及MFM 移动轨迹对移动波的吸引力。 本文的主要结果是:(a) 我们证明, 作为 美元( recentrent) 状态的固定性分布集中在一个( recentric) 流动波上( recentral) 。 (wec- crental) 也是( we- crental) a presental) a presental ( we- presental) a presental) a state) (web) be real) be sal) a real) a pres- be pres- be real) affactal) a pres( practal) is.