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- W2215771834 abstract "This paper concerns the Vertex reinforced jump process (VRJP) and the Edge reinforced random walk (ERRW) and their link with a random Schrodinger operator. On infinite graphs, we define a 1-dependent random potential $beta$ extending that defined in [15] on finite graphs, and consider its associated random Schrodinger 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 Schrodinger operator with eigenvalue $0$, when the VRJP is transient. Then we prove a representation of the VRJP as a mixture of Markov jump processes involving the function $psi$, the Green function of the random Schrodinger 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 VRJP and ERRW and a functional central limit theorem for VRJP and ERRW at weak reinforcement in dimension $dge 3$, using estimates of [9,7]. We also deduce recurrence of the ERRW in dimension $d=2$ for any initial constant weights, using the estimates of Merkl and Rolles, [12]. We conjecture some links between recurrence/transience of the VRJP and localization/delocalization of the random Schrodinger operator $H_beta$." @default.
- W2215771834 created "2016-06-24" @default.
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- W2215771834 date "2015-07-28" @default.
- W2215771834 modified "2023-09-27" @default.
- W2215771834 title "A random Schrodinger operator associated with the Vertex Reinforced Jump Process and the Edge Reinforced Random Walk" @default.
- W2215771834 hasPublicationYear "2015" @default.
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