MEAN FIELD BEHAVIOR DURING THE BIG BANG REGIME FOR COALESCING RANDOM WALKS

• 成果类型: Article
• 署名作者: Hermon, Jonathan; Li, Shuangping; Yao, Dong; Zhang, Lingfu
• 署名单位: University of British Columbia; Princeton University; Jiangsu Normal University; Princeton University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1571
• 发表日期: 2022
• 页码: 1813-1884
• 关键词: model; time

摘要

In this paper, we consider coalescing random walks on a general connected graph G = (V, E). We set up a unified framework to study the leading order of the decay rate of P-t , the expectation of the fraction of occupied sites at time t, particularly for the 'Big Bang' regime where t << t(coal) :=E(inf{s : There is only one particle at time s}). Our results show that P-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, (d) over bar] for some (d) over bar >= 3, and (2) finite and infinite transitive graphs. In the first case, we show that for 1 << t << vertical bar V vertical bar, P-t decays in the order of t(-1) , and (tP(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 (tP(t))(-1) is also roughly vertical bar V vertical bar/(2t(meet)), where t(meet) is the mean meeting time of two independent walkers. By taking the local weak limit, we prove convergence of tp(t) as t -> infinity for the corre- sponding unimodular Galton-Watson tree. For the second family of graphs, we consider a growing sequence of finite transitive graphs G(n) = (V-n, E-n), satisfying that t(meet) = O (vertical bar V-n vertical bar) and the inverse of the spectral gap trel is o(vertical bar V-n vertical bar). We show that t(rel) << t << t(coal), (tP(t))(-1) is approximately the prob- ability that two random walks never meet before time t, and it is also roughly vertical bar V vertical bar/(2t(meet)). In addition, we define a natural 'uniform transience' condition, and show that in the transitive setup it implies the above estimates of t P-t for all 1 << t << t(coal). Estimates of th are also obtained for all infinite transient transitive unimodular graphs, in particular, all transient transitive amenable graphs.