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Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in1986, 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 berepresented in terms of a Vertex-reinforced jump process (VRJP) withindependent gamma conductances: the VRJP was conceived by Werner and firststudied by Davis and Volkov (2002,2004), and is a continuous-time processfavouring sites with more local time. We calculate, for any finite graph $G$,the limiting measure of the centred occupation time measure of VRJP, andinterpret it as a supersymmetric hyperbolic sigma model in quantum fieldtheory, introduced by Zirnbauer (1991). This enables us to deduce that VRJP andERRW are positive recurrent in any dimension for large reinforcement, and thatVRJP is transient in dimension greater than or equal to 3 for smallreinforcement, using results of Disertori and Spencer (2010), Disertori,Spencer and Zirnbauer (2010).

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