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 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 , 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 .
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 \mathbb Z^d , d \geq 3, for small reinforcement, using results of Disertori and Spencer  and Disertori, Spencer and Zirnbauer .
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