Continuous-time vertex reinforced jump processes on Galton-Watson trees

TL;DR

This paper analyzes a continuous-time vertex reinforced jump process on a supercritical Galton-Watson tree, establishing a sharp phase transition at a critical parameter between recurrence and transience.

Contribution

It completes previous results by proving the existence and explicit value of a critical parameter that determines recurrence or transience.

Findings

Existence of a unique critical parameter c_crit

Recurrent for c ≤ c_crit, transient for c > c_crit

Explicit characterization of the phase transition

Abstract

We consider a continuous-time vertex reinforced jump process on a supercritical Galton-Watson tree. This process takes values in the set of vertices of the tree and jumps to a neighboring vertex with rate proportional to the local time at that vertex plus a constant $c$. The walk is either transient or recurrent depending on this parameter $c$. In this paper, we complete results previously obtained by Davis and Volkov [Probab. Theory Related Fields 123 (2002) 281-300, Probab. Theory Related Fields 128 (2004) 42-62] and Collevecchio [Ann. Probab. 34 (2006) 870-878, Electron. J. Probab. 14 (2009) 1936-1962] by proving that there is a unique (explicit) positive $c_{\mathrm{crit}}$ such that the walk is recurrent for $c\leq c_{\mathrm{crit}}$ and transient for $c>c_{\mathrm{crit}}$.