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：我们研究Cooper、McDowell、Radzik、Rivera与Shiraga（2018）提出的图上粒子同步运动过程。初始时刻，M个粒子置于图G的某个顶点。此后每个时间步，若某顶点存在至少两个粒子，则该顶点上的所有粒子独立随机跳向均匀选取的邻接顶点。当首次出现所有顶点至多含一个粒子时过程终止，我们将该（随机）时间步称为扩散时间。本文研究G为n阶完全图且粒子数M=n/2+αn^{1/2}+o(n^{1/2})（α∈ℝ）的情形。此M选取对应扩散时间过程的临界窗口。我们证明：经n^{-1/2}尺度变换后，扩散时间在p阶矩意义下（n→∞，p∈ℝ）收敛于连续且几乎必然为正的随机变量T_α。我们发现T_α等价于Lambert（2005）深入研究的标准逻辑分支过程的吸收时间，并给出其期望表达式。特别在临界窗口中心，我们证明𝔼[T_0]=π^{3/2}/√7，并给出当|α|充分大时量化进出临界窗口的显式渐近表达式。同时研究从初始到扩散时间终止前粒子总跳跃次数的计数随机变量，证明经n ln n尺度变换后该变量依概率收敛于2/7。