## 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.

## Funding Statement

SL and LZ were also provided 2019 AMS-MRC Collaboration Travel Funds, which enabled them a visit to Duke University, during which part of this work was completed. Both the workshop and the travel funds are supported by the National Science Foundation under Grant Number NSF DMS-1641020. JH is supported by NSERC grants.

## Acknowledgments

The authors are grateful to Rick Durrett for introducing the problem and sharing his perspectives. We also thank anonymous referees for carefully reading this manuscript and providing many useful comments and suggestions. The project was initiated in the workshop 2019 AMS-MRC: Stochastic Spatial Models.

## Citation

Jonathan Hermon. Shuangping Li. Dong Yao. Lingfu Zhang. "Mean field behavior during the Big Bang regime for coalescing random walks." Ann. Probab. 50 (5) 1813 - 1884, September 2022. https://doi.org/10.1214/22-AOP1571

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