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Transience of Edge-Reinforced Random Walk

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Abstract

We show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension \({d\ge3}\). 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 supersymmetric (SuSy) hyperbolic model for fixed conductances by Disertori et al. (Commun Math Phys 300:435–486, 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 SuSy hyperbolic sigma model developed by Sabot and Tarrès (J Eur Math Soc, arXiv:1111.3991, 2015).

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Authors and Affiliations

  1. Institute for Applied Mathematics and Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Margherita Disertori

  2. CNRS UMR 5208, Institut Camille Jordan, Université Lyon 1, Université de Lyon, 43, Boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Christophe Sabot

  3. Ceremade, CNRS UMR 7534 and Université Paris-Dauphine, Place de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Pierre Tarrès

Authors
  1. Margherita Disertori
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  2. Christophe Sabot
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  3. Pierre Tarrès
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Correspondence to Margherita Disertori.

Additional information

Communicated by H. Spohn

This work was supported by the ANR project MEMEMO2, ERC Starting Grant CoMBoS and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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Disertori, M., Sabot, C. & Tarrès, P. Transience of Edge-Reinforced Random Walk. Commun. Math. Phys. 339, 121–148 (2015). https://doi.org/10.1007/s00220-015-2392-y

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  • DOI: https://doi.org/10.1007/s00220-015-2392-y

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