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 variable counting the total number of jumps that are performed by the particles until the dispersion time is reached and prove that, if rescaled by $n\ln n$, it converges to $2/7$ in probability.

翻译：我们考虑 Cooper、McDowell、Radzik、Rivera 和 Shiraga（2018）引入的粒子在图 $G$ 顶点上同步移动过程。初始时，$M$ 个粒子置于 $G$ 的一个顶点上。在后续时间步中，所有位于至少有两个粒子占据的顶点上的粒子，独立地随机均匀跳向一个邻居。当首次出现无顶点被多于一个粒子占据时，过程终止；我们称此（随机）时间步为分散时间。本文研究 $G$ 为 $n$ 个顶点的完全图且粒子数 $M=n/2+\alpha n^{1/2} + o(n^{1/2})$（$\alpha\in \mathbb{R}$）的情形。此 $M$ 选择对应过程关于分散时间的临界窗口。我们证明：经 $n^{-1/2}$ 尺度缩放后，当 $n\rightarrow \infty$ 时，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 $T_\alpha$ 是标准逻辑分支过程的吸收时间（由 Lambert（2005）深入研究），并确定其期望值。特别地，在临界窗口中心，我们证明 $\mathbb{E}[T_0] = \pi^{3/2}/\sqrt{7}$，并进一步给出 $|\alpha|$ 较大时的显式渐近形式，以量化进入和离开临界窗口的过渡。我们还研究了计数直至分散时间达到时粒子执行总跳跃次数的随机变量，并证明经 $n\ln n$ 尺度缩放后，它依概率收敛于 $2/7$。