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arXiv · 1403.6079

Transience of Edge-Reinforced Random Walk

Margherita Disertori·Christophe Sabot·Pierre Tarrès
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Abstract

We show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension d greater than 2. This proves the existence of a phase transition between recurrent and transient behavior, thus solving an open problem stated by Diaconis in 1986. The argument adapts the proof of quasi-diffusive behavior of the SuSy hyperbolic model for fixed conductances by Disertori, Spencer and Zirnbauer [CMP 2010], using the representation of ERRW as a mixture of vertex-reinforced jump processes (VRJP) with independent gamma conductances, and the interpretation of the limit law of VRJP as a supersymmetric (SuSy) hyperbolic sigma model developed by Sabot and Tarrès in [JEMS 2014].

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Margherita Disertori, Christophe Sabot, Pierre Tarrès. 2014-08-30. Transience of Edge-Reinforced Random Walk. https://arxiv.org/abs/1403.6079

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