Mean field behavior during the Big Bang regime for coalescing random walks

Abstract

In this paper, we consider coalescing random walks on a general connected graph $\mathcal{G}=(\mathit{V},\mathit{E})$. We set up a unified framework to study the leading order of the decay rate of ${\mathit{P}}_{\mathit{t}}$, the expectation of the fraction of occupied sites at time t, particularly for the ‘Big Bang’ regime where $\mathit{t}\ll {\mathit{t}}_{\mathrm{coal}}:=\mathbb{E}(inf\{\mathit{s}:\text{There is only one particle at time}\mathit{s}\})$. Our results show that ${\mathit{P}}_{\mathit{t}}$ satisfies certain ‘mean-field behaviors’ if the graphs satisfy certain ‘transience-like’ conditions.

We apply this framework to two families of graphs: (1) graphs given by the configuration model whose degree distribution is supported in the interval $[3,\overline{\mathit{d}}]$ for some $\overline{\mathit{d}}\ge 3$, and (2) finite and infinite transitive graphs. In the first case, we show that for $1\ll \mathit{t}\ll |\mathit{V}|$, ${\mathit{P}}_{\mathit{t}}$ decays in the order of ${\mathit{t}}^{-1}$, and ${(\mathit{t}{\mathit{P}}_{\mathit{t}})}^{-1}$ is approximately 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. The number ${(\mathit{t}{\mathit{P}}_{\mathit{t}})}^{-1}$ is also roughly $|\mathit{V}|/(2{\mathit{t}}_{\text{meet}})$, where ${\mathit{t}}_{\text{meet}}$ is the mean meeting time of two independent walkers. By taking the local weak limit, we prove convergence of $\mathit{t}{\mathit{P}}_{\mathit{t}}$ as $\mathit{t}\to \infty$ for the corresponding unimodular Galton–Watson tree. For the second family of graphs, we consider a growing sequence of finite transitive graphs ${\mathcal{G}}_{\mathit{n}}=({\mathit{V}}_{\mathit{n}},{\mathit{E}}_{\mathit{n}})$, satisfying that ${\mathit{t}}_{\text{meet}}=\mathit{O}(|{\mathit{V}}_{\mathit{n}}|)$ and the inverse of the spectral gap ${\mathit{t}}_{\text{rel}}$ is $\mathit{o}(|{\mathit{V}}_{\mathit{n}}|)$. We show that ${\mathit{t}}_{\text{rel}}\ll \mathit{t}\ll {\mathit{t}}_{\text{coal}}$, ${(\mathit{t}{\mathit{P}}_{\mathit{t}})}^{-1}$ is approximately the probability that two random walks never meet before time t, and it is also roughly $|\mathit{V}|/(2{\mathit{t}}_{\text{meet}})$. In addition, we define a natural ‘uniform transience’ condition, and show that in the transitive setup it implies the above estimates of $\mathit{t}{\mathit{P}}_{\mathit{t}}$ for all $1\ll \mathit{t}\ll {\mathit{t}}_{\text{coal}}$. Estimates of $\mathit{t}{\mathit{P}}_{\mathit{t}}$ are also obtained for all infinite transient transitive unimodular graphs, in particular, all transient transitive amenable graphs.