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

Authors: Christophe Sabot and Xiaolin Zeng
Journal: J. Amer. Math. Soc. 32 (2019), 311-349
MSC (2010): Primary 60K35, 60K37; Secondary 82B44, 81T25, 81T60
Published electronically: August 16, 2018
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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 $.

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Additional Information

Christophe Sabot
Affiliation: Université de Lyon, Université Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43, Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France

Xiaolin Zeng
Affiliation: 108 Schreiber Building, School of Mathematics, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv 69978, Israel

Received by editor(s): January 20, 2016
Received by editor(s) in revised form: May 26, 2017, August 4, 2017, and June 5, 2018
Published electronically: August 16, 2018
Additional Notes: This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and by the ANR/FNS project MALIN (ANR-16-CE93-0003). The second author is supported by ERC Starting Grant 678520.
Article copyright: © Copyright 2018 American Mathematical Society