A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs

Abstract: 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 ${{\mathbb Z}}<sup>d$, 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 $H_\beta$.