We consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle; we call this (random) time step the dispersion time. In this work we study the case where $G$ is the complete graph on $n$ vertices and the number of particles is $M=n/2+\alpha n^{1/2} + o(n^{1/2})$, $\alpha\in \mathbb{R}$. This choice of $M$ corresponds to the critical window of the process, with respect to the dispersion time. We show that the dispersion time, if rescaled by $n^{-1/2}$, converges in $p$-th mean, as $n\rightarrow \infty$ and for any $p \in \mathbb{R}$, to a continuous and almost surely positive random variable $T_\alpha$. We find that $T_\alpha$ is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that $\mathbb{E}[T_0] = \pi^{3/2}/\sqrt{7}$, and furthermore we formulate explicit asymptotics when $|\alpha|$ gets large that quantify the transition into and out of the critical window. We also study the (random) total number of jumps that are performed by the particles until the dispersion time is reached. In particular, we prove that it centers around $\frac{2}{7}n\ln n$ and that it has variations linear in $n$, whose distribution we can describe explicitly.

翻译：我们考虑Cooper、McDowell、Radzik、Rivera与Shiraga（2018）提出的图$G$顶点上粒子同步运动过程。初始时刻，$M$个粒子置于$G$的某一顶点上。在每个后续时间步中，位于至少有两个粒子占据的顶点上的所有粒子独立地随机跳向均匀选择的邻居。该过程在首次无顶点被多于一个粒子占据时终止；我们称此（随机）时间步为分散时间。本文研究$G$为$n$顶点完全图且粒子数$M=n/2+\alpha n^{1/2} + o(n^{1/2})$（$\alpha\in \mathbb{R}$）的情形。此$M$的选取对应于过程关于分散时间的临界窗口。我们证明，当$n\rightarrow \infty$时，经$n^{-1/2}$尺度放缩的分散时间对任意$p \in \mathbb{R}$依$p$阶矩收敛于一个连续且几乎必然为正的随机变量$T_\alpha$。我们发现$T_\alpha$是Lambert（2005）深入研究的标准逻辑分支过程的吸收时间，并确定了其期望值。具体而言，在临界窗口中心处得到$\mathbb{E}[T_0] = \pi^{3/2}/\sqrt{7}$，并进一步给出了$|\alpha|$较大时量化临界窗口进出过渡的显式渐近表达式。我们还研究了过程达到分散时间前粒子执行的总跳跃次数（随机变量）。特别地，我们证明其中心项为$\frac{2}{7}n\ln n$，且存在线性阶的$n$的波动，其分布可显式刻画。