Ann. Probab. 50 (5), 1813-1884, (September 2022) DOI: 10.1214/22-AOP1571
Jonathan Hermon, Shuangping Li, Dong Yao, Lingfu Zhang
KEYWORDS: Coalescing random walks, Kingman’s coalescent, 60J90, 60J27
In this paper, we consider coalescing random walks on a general connected graph . We set up a unified framework to study the leading order of the decay rate of , the expectation of the fraction of occupied sites at time t, particularly for the ‘Big Bang’ regime where . Our results show that 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 for some , and (2) finite and infinite transitive graphs. In the first case, we show that for , decays in the order of , and 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 is also roughly , where is the mean meeting time of two independent walkers. By taking the local weak limit, we prove convergence of as for the corresponding unimodular Galton–Watson tree. For the second family of graphs, we consider a growing sequence of finite transitive graphs , satisfying that and the inverse of the spectral gap is . We show that , is approximately the probability that two random walks never meet before time t, and it is also roughly . In addition, we define a natural ‘uniform transience’ condition, and show that in the transitive setup it implies the above estimates of for all . Estimates of are also obtained for all infinite transient transitive unimodular graphs, in particular, all transient transitive amenable graphs.