Localization for Linearly Edge Reinforced Random Walks

Abstract

We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on non-amenable graphs. While we rely on the description of the LRRW as a mixture of Markov chains, the proof does not use the magic formula. We also derive analogous results for the vertex reinforced jump process.

30 pages

References in corpus (3)

• A survey of random processes with reinforcement
• Recurrence of edge-reinforced random walk on a two-dimensional graph
• Anderson localization for a supersymmetric sigma model

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• Invariance principles for random walks in random environment on trees
• Recurrence of horizontal-vertical walks
• Vertex-reinforced jump process on the integers with nonlinear reinforcement
• Two repelling random walks on $\mathbb{Z}$
• Transience of vertex-reinforced jump processes with long-range jumps
• Random walkers on a deformable medium