Edge-reinforced random walk, Vertex-Reinforced Jump Process and the supersymmetric hyperbolic sigma model
Sabot, Christophe; Tarres, Pierre (2015), Edge-reinforced random walk, Vertex-Reinforced Jump Process and the supersymmetric hyperbolic sigma model, Journal of the European Mathematical Society, 17, 9, p. 2353–2378. 10.4171/JEMS/559
TypeArticle accepté pour publication ou publié
Journal nameJournal of the European Mathematical Society
MetadataShow full item record
Institut Camille Jordan [Villeurbanne] [ICJ]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)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).
Subjects / KeywordsSigma model; Vertex Reinforced Random Walks; Edge Reinforced Random Walks
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