Site recurrence for coalescing random walk. (English) Zbl 1345.60110

Benjamini, Itai; Foxall, Eric; Gurel-Gurevich, Ori; Junge, Matthew; Kesten, Harry

Electron. Commun. Probab. 21, Paper No. 47, 12 p. (2016).

Summary: Begin continuous-time random walks from every vertex of a graph and have particles coalesce when they collide. We use a duality relation with the voter model to prove that the process is site recurrent on bounded degree graphs, and for Galton-Watson trees whose offspring distribution has exponential tail. We prove bounds on the occupation probability of a site, as well as a general 0-1 law. Similar conclusions hold for a coalescing process on trees where particles do not backtrack.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G50	Sums of independent random variables; random walks
60J27	Continuous-time Markov processes on discrete state spaces
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60F20	Zero-one laws

Keywords:

coalescing random walks; interacting particle system; site recurrence; Galton-Watson trees; zero-one law