Mean Field Behavior during the Big Bang for Coalescing Random Walk
In this paper we consider the coalescing random walk model on general graphs $\mathcal{G}=(V,E)$.
We set up a unified framework to study the leading order of decay rate of $P_t$, the fraction of occupied sites at time $t$, and we are particularly interested in the 'Big Bang' regime, where $t \ll t_{\mathrm{coal}}:=\mathbb{E}[\inf\{s:\text{There is only one particle at time }s \}]$.
Our results show that $P_t$ satisfies certain 'mean field behavior', if the graphs satisfy a certain 'transience-like' condition.
We apply this framework to two families of graphs: (1) graphs given by the configuration model with minimal degree at least 3, and (2) vertex-transitive graphs.
In the first case, we show that for $1 \ll t \ll |V|$, $P_t$ decays in the order of $t^{-1}$.
We also determine the constant by showing that $tP_t$ is close to the probability that two particles starting from the root of the corresponding unimodular Galton-Watson tree never collide after one of them leaves the root. For the second family of graphs, taking a growing sequence of finite vertex-transitive graphs $\mathcal{G}_n=(V_n, E_n)$ such that the mean meeting time $t_{\mathrm{meet}}$ is $O(|V_n|)$ and the inverse of the spectral gap $t_{\mathrm{rel}}$ is $o(|V_n|)$, we show that for
$t_{\mathrm{rel}} \ll t \ll t_{\mathrm{coal}}$, \begin{equation*} \begin{split} tP_t & = (1 \pm o(1))/(\mathbb{P}(\mbox{two random walks never meet before time }t))\\ &= (2+o(1))t_{\mathrm{meet}}/|V_n|. \end{split} \end{equation*} The first equality is also shown to hold for all infinite transient transitive unimodular graphs, in particular, all transient transitive amenable graphs. In addition, we define a certain natural `uniform transience' condition for a sequence of finite graphs, and show that in the transitive setup it implies the above for all $1 \ll t\ll t_{\mathrm{coal}}$.
Based on joint work with Jonathan Hermon, Shuangping Li and Lingfu Zhang.