About the asymptotic behaviour of the martingale associated with the Vertex Reinforced Jump Process on trees and Z d

TL;DR

This paper analyzes the long-term behavior of a martingale linked to the Vertex Reinforced Jump Process on trees and Z^d, revealing boundedness, moment estimates, and asymptotic rates across different phases.

Contribution

It provides new asymptotic results and moment estimates for the VRJP martingale, including explicit rates and behavior at critical points, using branching random walk techniques.

Findings

Martingale is bounded in L^p for p>1 on trees.

Explicit asymptotic rates for the martingale in recurrent and critical phases.

VRJP process at criticality is a mixture of positive recurrent Markov chains.

Abstract

We study the asymptotic behaviour of the martingale ($\psi$ n (o)) n$\in$N associated with the Vertex Reinforced Jump Process (VRJP). We show that it is bounded in L p for every p > 1 on trees and uniformly integrable on Z d in all the transient phase of the VRJP. Moreover, when the VRJP is recurrent on trees, we have good estimates on the moments of $\psi$ n (o) and we can compute the exact decreasing rate $\tau$ such that n --1 ln($\psi$ n (o)) $\sim$ --$\tau$ almost surely where $\tau$ is related to standard quantities for branching random walks. Besides, on trees, at the critical point, we show that n --1/3 ln($\psi$ n (o)) $\sim$ --$\rho$ c almost surely where $\rho$ c can be computed explicitely. Furthermore, at the critical point, we prove that the discrete process associated with the VRJP is a mixture of positive recurrent Markov chains. Our proofs use properties of the $\beta$-potential associated with the VRJP and techniques coming from the domain of branching random walks.