Edge-reinforced random walk, Vertex-Reinforced Jump Process and the supersymmetric hyperbolic sigma modelReport as inadecuate




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1 ICJ - Institut Camille Jordan Villeurbanne 2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

Abstract : Edge-reinforced random walk ERRW, introduced by Coppersmith and Diaconis in 1986, is a random process that takes values in the vertex set of a graph G, which is more likely to cross edges it has visited before. We show that it can be interpreted as an annealed version of the Vertex-reinforced jump process VRJP, conceived by Werner and first studied by Davis and Volkov 2002,2004, a continuous-time process favouring sites with more local time. We calculate, for any finite graph G, the limiting measure of the centred occupation time measure of VRJP, and interpret it as a supersymmetric hyperbolic sigma model in quantum field theory. This enables us to deduce that VRJP is recurrent in any dimension for large reinforcement, using a localisation result of Disertori and Spencer 2010.

Keywords : Sigma model Vertex Reinforced Random Walks Edge Reinforced Random Walks





Author: Christophe Sabot - Pierre Tarrès -

Source: https://hal.archives-ouvertes.fr/



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