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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs
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by Christophe Sabot and Xiaolin Zeng HTML | PDF
J. Amer. Math. Soc. 32 (2019), 311-349 Request permission

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
  • MR Author ID: 600825
  • Email: sabot@math.univ-lyon1.fr
  • Xiaolin Zeng
  • Affiliation: 108 Schreiber Building, School of Mathematics, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv 69978, Israel
  • MR Author ID: 1171462
  • Email: xzeng@math.univ-lyon1.fr
  • 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.
  • © Copyright 2018 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 32 (2019), 311-349
  • MSC (2010): Primary 60K35, 60K37; Secondary 82B44, 81T25, 81T60
  • DOI: https://doi.org/10.1090/jams/906
  • MathSciNet review: 3904155