Speed of Vertex reinforced jump process on Galton-Watson trees

TL;DR

This paper provides an alternative proof of the phase transition in recurrence and transience for the vertex reinforced jump process on Galton-Watson trees and investigates the transition between positive and null speed in the transient regime.

Contribution

It offers a new proof of the phase transition in recurrence and transience and analyzes the speed transition in the associated discrete-time process.

Findings

Phase transition between recurrence and transience as a function of initial local time c.

Existence of a phase transition between positive and null speed in the transient regime.

Application of techniques to establish speed behavior in the process.

Abstract

We give an alternative proof of the fact that the vertex reinforced jump process on Galton- Watson tree has a phase transition between recurrence and transience as a function of c, the initial local time, see [3]. Further, applying the techniques in [1], we show a phase transition between positive speed and null speed for the discrete time process associated in the transient regime.