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 \emph{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.

based on joint work with umberto de ambroggio, tamás makai, and annika steibel; see arxiv:2403.05372