A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs

Abstract

This paper concerns the Vertex reinforced jump process (VRJP), the Edge reinforced random walk (ERRW) and their link with a random Schrödinger operator. On infinite graphs, we define a 1-dependent random potential β extending that defined in [20] on finite graphs, and consider its associated random Schrödinger operator H β . We construct a random function ψ as a limit of martingales, such that ψ = 0 when the VRJP is recurrent, and ψ is a positive generalized eigenfunction of the random Schrödinger operator with eigenvalue 0, when the VRJP is transient. Then we prove a representation of the VRJP on infinite graphs as a mixture of Markov jump processes involving the function ψ, the Green function of the random Schrödinger operator and an independent Gamma random variable. On Z d , we deduce from this representation a zero-one law for recurrence or transience of the VRJP and the ERRW, and a functional central limit theorem for the VRJP and the ERRW at weak reinforcement in dimension d ≥ 3, using estimates of [10,8]. Finally, we deduce recurrence of the ERRW in dimension d = 2 for any initial constant weights (using the estimates of Merkl and Rolles, [15,17]), thus giving a full answer to the old question of Diaconis. We also raise some questions on the links between recurrence/transience of the VRJP and localization/delocalization of the random Schrödinger operator H β .

The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees

“…We should mention that a similar phase transition was conjectured for linearly edge-reinforced random walks (ERRW) on Z d in the eighties [7], and was first proved on regular trees in [16]. Only recently, the phase transition recurrence/transience on Z d , d ≥ 3, was established in [18,1,9], see also [19]. However, techniques developed for ERRW do not apply to ORRW, in particular because exchangeability does not hold.…”

Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

“…Consistency of the law of β was first observed by Sabot and Zeng in [SZ15]; see also Lemma 2.4 in [DMR17]. By Kolmogorov's consistency theorem, there is a probability space…”

“…More recently Sabot, Tarrès, and Zeng developed further the random Schrödinger operator interpretation [STZ17] [SZ15]. In particular they derived the explicit law for the random potential, and constructed two families of martingales in discrete time.…”

Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

“…See [1] for a discussion and precise statements. It has also been shown that the linearly edge-reinforced random walk (ERRW) with constant initial weights is recurrent in two dimensions [19,24], but the recurrence of the VRJP for all initial rates was an open problem until the present work. The relation between the ERRW and VRJP is discussed below.…”

“…The analogue of Theorem 1.1 for the ERRW with constant initial weights was established in [19,24], but not for the VRJP. Mermin-Wagner type theorems have also been proven for the ERRW in one and two dimensions [18,19].…”