报告题目：Mean Field Behavior during the Big Bang Regime for Coalescing Random Walk

报告人：姚东 Duke University           

报告时间：6月18周五10：00-11：00

报告地点：管理楼1418

摘要：

The talk is concerned with the coalescing random walk model on general graphs G=(V,E). Initially every vertex of G has a particle. Each particle performs independent random walk. Whenever two particles meet, they merge into one particle which continues to perform random walk. We set up a unified framework to study the leading order of 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 left at time s}].

Our results show that P_t satisfies certain `mean field behavior', if the graphs satisfy certain ‘transience-like’ conditions. We apply this framework to two families of graphs: (1) graphs generated by configuration model with degree at least 3, and (2) finite and infinite vertex-transitive graphs. In the first case, we show that for t in the Big Bang regime, (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. In the second case we establish similar results for finite ‘uniformly transient’ graphs and infinite transient transitive unimodular graphs. This answers a question due to Durrett.

Joint work with Jonathan Hermon (UBC), Shuangping Li (Princeton) and Lingfu Zhang (Princeton).