# Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model

### Christophe Sabot

Université Lyon 1, Villeurbanne, France### Pierre Tarrès

CNRS and Université Paris-Dauphine, France

## Abstract

Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986 [8], 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 [10, 11], 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 in 1991 [35].

This enables us to deduce that VRJP and ERRWare positive recurrent on any graph of bounded degree for large reinforcement, and that the VRJP is transient on $Z_{d},d≥3$, for small reinforcement, using results of Disertori and Spencer [15] and Disertori, Spencer and Zirnbauer [16].

## Cite this article

Christophe Sabot, Pierre Tarrès, Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. 17 (2015), no. 9, pp. 2353–2378

DOI 10.4171/JEMS/559