Reinforced random walks and statistical physics

In a joint work with Christophe Sabot (2011), we show an explicit link between the Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, and the supersymmetric hyperbolic sigma model developed by Zirnbauer in 1996, through another continuous-time process conceived by Werner in 2000, the Vertex-reinforced jump process (VRJP), favouring sites with more local time.

This correspondence enables to show Diaconis' conjecture that ERRW is strongly recurrent in any dimension for large reinforcement (in fact on any graph of bounded degree), and that it is transient in dimension larger than 3, using localisation/delocalization techniques of Disertori, Spencer and Zirnbauer (2010).