Edge-reinforced random walk on a ladder

Abstract

We prove that the edge-reinforced random walk on the ladder ${\mathbb{Z}\times\{1,2\}}$ with initial weights $a>3/4$ is recurrent. The proof uses a known representation of the edge-reinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a marginal of a multicomponent Gibbsian process. A transfer operator technique and entropy estimates from statistical mechanics are used to analyze this Gibbsian process. Furthermore, we prove spatially exponentially fast decreasing bounds for normalized local times of the edge-reinforced random walk on a finite piece of the ladder, uniformly in the size of the finite piece.Comment: Published at http://dx.doi.org/10.1214/009117905000000396 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A random environment for linearly edge-reinforced random walks on infinite graphs

“…This description was first given by Coppersmith and Diaconis in [CD86] and refined by Keane and Rolles in [KR00]. It has already been useful to analyze edge-reinforced random walk on certain infinite graphs, including ladders of arbitrary width, by taking the infinite volume limit of finite subgraphs; see [MR05b], [Rol05], and [MR05a]. However, in this paper, we do not make use of the explicit form of the mixing measure for finite graphs.…”

Linearly edge-reinforced random walks

“…Only recently, recurrence results were obtained. The following result was proved in [22]. The result was extended in [29], under the assumption the initial weight a is large: Theorem 7.2.…”

Edge-reinforced random walk on one-dimensional periodic graphs

“…The later development indeed followed this idea: In [5], it was shown that the edge-reinforced random walk on the ladder Z × {1, 2} with all initial weights equal to the same constant b > 3/4 is recurrent. For the edge-reinforced random walk on Z×T with a finite tree T and constant large initial weights, recurrence was shown in [11].…”

“…The techniques used below are different from the methods in [5,7,11]. To analyze the edge-reinforced random walk on the infinite periodic graph, we first study the process on finite periodic graphs.…”

“…The distribution of the random environment is explicitly known. In [5,7,11], this random environment was studied using a transfer operator method, which was technically quite involved due to the complicated dependence structure of the random environment. In the present paper, we pursue a different approach.…”