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Localization for Linearly Edge Reinforced Random Walks
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.Comment: 30 page
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