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

We study the asymptotic behaviour of the martingale (ψ n (o)) n∈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 ψ n (o) and we can compute the exact decreasing rate τ such that n −1 ln(ψ n (o)) ∼ −τ almost surely where τ is related to standard quantities for branching random walks. Besides, on trees, at the critical point, we show that n −1/3 ln(ψ n (o)) ∼ −ρ c almost surely where ρ 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 β-potential associated with the VRJP and techniques coming from the domain of branching random walks