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 $G$. At the beginning of each time step, for every vertex inhabited by at least two particles, each of these particles moves 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. Cooper et al. showed that when the underlying graph is the complete graph on~$n$ vertices, then there is a phase transition when the number of particles $M = n/2$. They showed that if $M<(1-\varepsilon)n/2$ for some fixed $\varepsilon>0$, then the process finishes in a logarithmic number of steps, while if $M>(1+\varepsilon)n/2$, an exponential number of steps are required with high probability. In this paper we provide a thorough analysis of the dispersion time around criticality, where $\varepsilon = o(1)$, and describe the fine details of the transition between logarithmic and exponential time. As a consequence of our results we establish, for example, that the dispersion time is in probability and in expectation $\Theta(n^{1/2})$ when $|\varepsilon| = O(n^{-1/2})$, and provide qualitative bounds for its tail behavior.

翻译：我们研究由Cooper、McDowell、Radzik、Rivera和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$G$的某个顶点上。在每个时间步开始时，对于至少有两个粒子的顶点，每个粒子独立地随机均匀移动到相邻顶点。当所有顶点至多含有一个粒子时，过程终止。Cooper等人证明：当底层图为$n$个顶点的完全图时，粒子数$M = n/2$处存在相变。他们指出，若对某固定$\varepsilon>0$有$M<(1-\varepsilon)n/2$，则过程在对数步数内结束；而若$M>(1+\varepsilon)n/2$，则高概率需要指数步数。本文在临界点附近（$\varepsilon = o(1)$）对分散时间进行细致分析，揭示了对数时间与指数时间之间过渡的精细结构。例如，我们证明当$|\varepsilon| = O(n^{-1/2})$时，分散时间的概率与期望均为$\Theta(n^{1/2})$，并给出其尾部行为的定性界限。