Duke Mathematical Journal

Localization for linearly edge reinforced random walks

Omer Angel, Nicholas Crawford, and Gady Kozma

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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 nonamenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on nonamenable graphs. While we rely on the equivalence of the LRRW to a mixture of Markov chains, the proof does not use the so-called magic formula which is central to most work on this model. We also derive analogous results for the vertex reinforced jump process.

Article information

Duke Math. J., Volume 163, Number 5 (2014), 889-921.

First available in Project Euclid: 26 March 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments


Angel, Omer; Crawford, Nicholas; Kozma, Gady. Localization for linearly edge reinforced random walks. Duke Math. J. 163 (2014), no. 5, 889--921. doi:10.1215/00127094-2644357. https://projecteuclid.org/euclid.dmj/1395856218

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