Recurrence of edge-reinforced random walk on a two-dimensional graph

Abstract

We consider a linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from $\mathbb{Z}^2$ by replacing every edge by a sufficiently large, but fixed number of edges in series. We prove that the linearly edge-reinforced random walk on these graphs is recurrent. Furthermore, we derive bounds for the probability that the edge-reinforced random walk hits the boundary of a large box before returning to its starting point.Comment: Published in at http://dx.doi.org/10.1214/08-AOP446 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

“…Finally, we can deduce recurrence of the ERRW in dimension 2 from Theorem 1, Proposition 5 and the estimates obtained by Merkl and Rolles in [15,17] 1 .…”

“…e.g. [1,8,17,20]. In particular, in was proved in 2012 by Sabot, Tarrès, [20], and Angel, Crawford, Kozma, [1], on any graph with bounded degree at strong reinforcement (i.e.…”

“…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 .…”

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.…”

“…for all β > 0 in the case of H 2 and for all sufficiently large β > 0 for H 2|2 . In the H 2|2 case (19) corresponds to transience of the VRJP (in the sense of bounded expected local time, see Corollary 1.10 below) and to the uniform boundedness (in the spectral parameter z ∈ C + ) of the expected square of the absolute value of the resolvent for random band matrices in the sigma model approximation [27] (recall Section 1.1). It also implies that the hyperbolic symmetry is spontaneously broken.…”