A random Schr\"odinger operator associated with the Vertex Reinforced Jump Process on infinite graphs

This paper concerns the Vertex reinforced jump process (VRJP), the Edge reinforced random walk (ERRW) and their link with a random Schr\"odinger operator. On infinite graphs, we define a 1-dependent random potential $\beta$ extending that defined in [20] on finite graphs, and consider its associated random Schr\"odinger operator $H_\beta$. We construct a random function $\psi$ as a limit of martingales, such that $\psi=0$ when the VRJP is recurrent, and $\psi$ is a positive generalized eigenfunction of the random Schr\"odinger operator with eigenvalue $0$, when the VRJP is transient. Then we prove a representation of the VRJP on infinite graphs as a mixture of Markov jump processes involving the function $\psi$, the Green function of the random Schr\"odinger operator and an independent Gamma random variable. On ${{\mathbb Z}}^d$, we deduce from this representation a zero-one law for recurrence or transience of the VRJP and the ERRW, and a functional central limit theorem for the VRJP and the ERRW at weak reinforcement in dimension $d\ge 3$, using estimates of [10,8]. Finally, we deduce recurrence of the ERRW in dimension $ d=2$ for any initial constant weights (using the estimates of Merkl and Rolles, [15,17]), thus giving a full answer to the old question of Diaconis. We also raise some questions on the links between recurrence/transience of the VRJP and localization/delocalization of the random Schr\"odinger operator $H_\beta$.