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

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Christophe Sabot and Xiaolin Zeng
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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 $H_\beta$.## References

- Omer Angel, Nicholas Crawford, and Gady Kozma,
*Localization for linearly edge reinforced random walks*, Duke Math. J.**163**(2014), no. 5, 889–921. MR**3189433**, DOI 10.1215/00127094-2644357 - Anne-Laure Basdevant and Arvind Singh,
*Continuous-time vertex reinforced jump processes on Galton-Watson trees*, Ann. Appl. Probab.**22**(2012), no. 4, 1728–1743. MR**2985176**, DOI 10.1214/11-AAP811 - Andrea Collevecchio,
*Limit theorems for vertex-reinforced jump processes on regular trees*, Electron. J. Probab.**14**(2009), no. 66, 1936–1962. MR**2540854**, DOI 10.1214/EJP.v14-693 - D. Coppersmith and P. Diaconis,
*Random walk with reinforcement*, unpublished manuscript, pages 187–220, 1987. - Burgess Davis and Stanislav Volkov,
*Vertex-reinforced jump processes on trees and finite graphs*, Probab. Theory Related Fields**128**(2004), no. 1, 42–62. MR**2027294**, DOI 10.1007/s00440-003-0286-y - A. De Masi, P. A. Ferrari, S. Goldstein, and W. D. Wick,
*An invariance principle for reversible Markov processes. Applications to random motions in random environments*, J. Statist. Phys.**55**(1989), no. 3-4, 787–855. MR**1003538**, DOI 10.1007/BF01041608 - Margherita Disertori, Franz Merkl, and Silke W. W. Rolles,
*A supersymmetric approach to martingales related to the vertex-reinforced jump process*, ALEA Lat. Am. J. Probab. Math. Stat.**14**(2017), no. 1, 529–555. MR**3663098**, DOI 10.30757/alea.v14-27 - Margherita Disertori, Christophe Sabot, and Pierre Tarrès,
*Transience of edge-reinforced random walk*, Comm. Math. Phys.**339**(2015), no. 1, 121–148. MR**3366053**, DOI 10.1007/s00220-015-2392-y - M. Disertori and T. Spencer,
*Anderson localization for a supersymmetric sigma model*, Comm. Math. Phys.**300**(2010), no. 3, 659–671. MR**2736958**, DOI 10.1007/s00220-010-1124-6 - M. Disertori, T. Spencer, and M. R. Zirnbauer,
*Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model*, Comm. Math. Phys.**300**(2010), no. 2, 435–486. MR**2728731**, DOI 10.1007/s00220-010-1117-5 - Paul R. Halmos,
*Measure Theory*, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR**0033869**, DOI 10.1007/978-1-4684-9440-2 - M. S. Keane and S. W. W. Rolles,
*Edge-reinforced random walk on finite graphs*, Infinite dimensional stochastic analysis (Amsterdam, 1999) Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., vol. 52, R. Neth. Acad. Arts Sci., Amsterdam, 2000, pp. 217–234. MR**1832379** - G. Letac,
*Personal communication*, 2015. - Russell Lyons and Yuval Peres,
*Probability on trees and networks*, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42, Cambridge University Press, New York, 2016. MR**3616205**, DOI 10.1017/9781316672815 - Franz Merkl and Silke W. W. Rolles,
*Bounding a random environment for two-dimensional edge-reinforced random walk*, Electron. J. Probab.**13**(2008), no. 19, 530–565. MR**2399290**, DOI 10.1214/EJP.v13-495 - Franz Merkl and Silke W. W. Rolles,
*A random environment for linearly edge-reinforced random walks on infinite graphs*, Probab. Theory Related Fields**138**(2007), no. 1-2, 157–176. MR**2288067**, DOI 10.1007/s00440-006-0016-3 - Franz Merkl and Silke W. W. Rolles,
*Recurrence of edge-reinforced random walk on a two-dimensional graph*, Ann. Probab.**37**(2009), no. 5, 1679–1714. MR**2561431**, DOI 10.1214/08-AOP446 - Robin Pemantle,
*Phase transition in reinforced random walk and RWRE on trees*, Ann. Probab.**16**(1988), no. 3, 1229–1241. MR**942765** - Abraham Berman and Robert J. Plemmons,
*Nonnegative matrices in the mathematical sciences*, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR**544666** - Christophe Sabot and Pierre Tarrès,
*Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model*, J. Eur. Math. Soc. (JEMS)**17**(2015), no. 9, 2353–2378. MR**3420510**, DOI 10.4171/JEMS/559 - Christophe Sabot and Pierre Tarres,
*Inverting Ray-Knight identity*, Probab. Theory Related Fields**165**(2016), no. 3-4, 559–580. MR**3520013**, DOI 10.1007/s00440-015-0640-x - Christophe Sabot, Pierre Tarrès, and Xiaolin Zeng,
*The vertex reinforced jump process and a random Schrödinger operator on finite graphs*, Ann. Probab.**45**(2017), no. 6A, 3967–3986. MR**3729620**, DOI 10.1214/16-AOP1155 - L. Tournier,
*A note on the recurrence of edge reinforced random walks*, arXiv preprint arXiv:0911.5255, 2009. - Martin R. Zirnbauer,
*Fourier analysis on a hyperbolic supermanifold with constant curvature*, Comm. Math. Phys.**141**(1991), no. 3, 503–522. MR**1134935**, DOI 10.1007/BF02102812

## 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