## Duke Mathematical Journal

### 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 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

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

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.dmj/1395856218

Digital Object Identifier
doi:10.1215/00127094-2644357

Mathematical Reviews number (MathSciNet)
MR3189433

Zentralblatt MATH identifier
1302.60129

#### Citation

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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