Friday, March 2, 2012

1111.3991 (Christophe Sabot et al.)

Edge-reinforced random walk, Vertex-Reinforced Jump Process and the
supersymmetric hyperbolic sigma model

Christophe Sabot, Pierre Tarres
Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process, which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is 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 and ERRW are recurrent in any dimension for large reinforcement, using a localisation result of Disertori and Spencer (2010).
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