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

Abstract

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, introduced by Zirnbauer (1991). This enables us to deduce that VRJP and ERRW are positive recurrent in any dimension for large reinforcement, and that VRJP is transient in dimension greater than or equal to 3 for small reinforcement, using results of Disertori and Spencer (2010), Disertori, Spencer and Zirnbauer (2010).

23 pages, 1 figure

References in corpus (7)

• A survey of random processes with reinforcement
• Bayesian analysis for reversible Markov chains
• Vertex-reinforced random walk on Z eventually gets stuck on five points
• Recurrence of edge-reinforced random walk on a two-dimensional graph
• Bayesian analysis of variable-order, reversible Markov chains
• Random Dirichlet environment viewed from the particle in dimension $d\ge 3$
• Localization of reinforced random walks

Cited by in corpus (2)

• Localization for a nonlinear sigma model in a strip related to vertex reinforced jump processes
• Monotone interaction of walk and graph: recurrence versus transience