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 relation to a random Schrödinger operator. On infinite graphs, we define a 1-dependent random potential $\beta$ extending that defined by Sabot, Tarrès, and Zeng on finite graphs, and consider its associated random Schrödinger 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 Schrödinger 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 Schrödinger operator, and an independent Gamma random variable. On ${\Bbb Z}^d$, 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 Disertori, Sabot, and Tarrès and of Disertori, Spencer, and Zimbauer. Finally, we deduce recurrence of the ERRW in dimension $d=2$ for any initial constant weights (using the estimates of Merkl and Rolles), thus giving a full answer to the question raised by Diaconis. We also raise some questions on the links between recurrence/transience of the VRJP and localization/delocalization of the random Schrödinger operator $H_\beta$.