On the recurrence of edge-reinforced random walk on ℤ×G

Abstract

Let G be a finite tree. It is shown that edge-reinforced random walk on Z × G with large initial weights is recurrent. This includes recurrence on multi-level ladders of arbitrary width. For edge-reinforced random walk on {0, 1, . . . , n} × G, it is proved that asymptotically, with high probability, the normalized edge local times decay exponentially in the distance from the starting level. The estimates are uniform in n. They are used in the recurrence proof.

Edge-reinforced random walk on one-dimensional periodic graphs

“…This simplifies considerably the present proof compared to the previous papers [5,7,11]. In those papers, the estimates for the contributions from the spanning trees were quite involved.…”

“…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]. In [7], a more detailed analysis of the edge-reinforced random walk with large constant initial weights on N 0 × {1, 2} and on N 0 × T with a finite tree T was given.…”

“…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.…”

Linearly edge-reinforced random walks

“…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. Let G be a finite tree.…”