Probability Theory and Related Fields

, Volume 173, Issue 3–4, pp 1349–1387 | Cite as

Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model

  • Franz Merkl
  • Silke W. W. RollesEmail author
  • Pierre Tarrès


In this paper, we define an extension of the supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer. We show that it arises as a weak joint limit of a time-changed version introduced by Sabot and Tarrès of the vertex-reinforced jump process. It describes the asymptotics of rescaled crossing numbers, rescaled fluctuations of local times, asymptotic local times on a logarithmic scale, endpoints of paths, and last exit trees.


Vertex-reinforced jump process Self-interacting random walks Supersymmetric hyperbolic nonlinear sigma model 

Mathematics Subject Classification

Primary 60K35 Secondary 81T60 



The authors would like to thank an anonymous referee and the associate editor for very constructive comments helping us to improve the paper.


  1. 1.
    Angel, O., Crawford, N., Kozma, G.: Localization for linearly edge reinforced random walks. Duke Math. J. 163(5), 889–921 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Basdevant, A.L., Singh, A.: Continuous-time vertex reinforced jump processes on Galton–Watson trees. Ann. Appl. Probab. 22(4), 1728–1743 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Collevecchio, A.: On the transience of processes defined on Galton–Watson trees. Ann. Probab. 34(3), 870–878 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Collevecchio, A.: Limit theorems for vertex-reinforced jump processes on regular trees. Electron. J. Probab. 14(66), 1936–1962 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Davis, B., Volkov, S.: Continuous time vertex-reinforced jump processes. Probab. Theory Relat. Fields 123(2), 281–300 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Davis, B., Volkov, S.: Vertex-reinforced jump processes on trees and finite graphs. Probab. Theory Relat. Fields 128(1), 42–62 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Disertori, M., Merkl, F., Rolles, S.: Localization for a nonlinear sigma model in a strip related to vertex reinforced jump processes. Commun. Math. Phys. 332(2), 783–825 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Disertori, M., Merkl, F., Rolles, S.: A comparison of a nonlinear sigma model with general pinning and pinning at one point. Electron. J. Probab. 21, 27, 16 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Disertori, M., Sabot, C., Tarrès, P.: Transience of edge-reinforced random walk. Commun. Math. Phys. 339(1), 121–148 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Disertori, M., Spencer, T.: Anderson localization for a supersymmetric sigma model. Commun. Math. Phys. 300(3), 659–671 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Disertori, M., Spencer, T., Zirnbauer, M.: Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model. Commun. Math. Phys. 300(2), 435–486 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Keane, M., Rolles, S.: Edge-reinforced random walk on finite graphs. In: Infinite Dimensional Stochastic Analysis (Amsterdam, 1999), pp. 217–234. Royal Netherlands Academy of Arts and Sciences, Amsterdam (2000)Google Scholar
  13. 13.
    Merkl, F., Rolles, S.: Linearly edge-reinforced random walks. In: Dynamics and Stochastics. Festschrift in Honor of M. S. Keane. Selected Papers Based on the Presentations at the Conference ‘Dynamical Systems, Probability Theory, and Statistical Mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the Occasion of the 65th Birthday of Mike S. Keane., pp. 66–77. IMS, Institute of Mathematical Statistics, Beachwood, OH (2006).
  14. 14.
    Sabot, C., Tarrès, P.: Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. JEMS 17(9), 2353–2378 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sabot, C., Tarrès, P.: Inverting Ray-Knight identity. Probab. Theory Relat. Fields 165(3–4), 559–580 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sabot, C., Tarrès, P., Zeng, X.: The vertex reinforced jump process and a random Schrödinger operator on finite graphs. Ann. Probab. 45(6A), 3967–3986 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sabot, C., Zeng, X.: A random Schrödinger operator associated with the vertex reinforced jump process and the edge reinforced random walk. Preprint arXiv:1507.07944 (2015)
  18. 18.
    Tarrès, P.: Localization of reinforced random walks. Preprint arXiv:1103.5536 (2011)
  19. 19.
    Zeng, X.: How vertex reinforced jump process arises naturally. Ann. Inst. Henri Poincaré Probab. Stat. 52(3), 1061–1075 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zirnbauer, M.: Fourier analysis on a hyperbolic supermanifold with constant curvature. Commun. Math. Phys. 141(3), 503–522 (1991).

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Franz Merkl
    • 1
  • Silke W. W. Rolles
    • 2
    Email author
  • Pierre Tarrès
    • 3
    • 4
  1. 1.Mathematical InstituteLudwig-Maximilians-Universität MünchenMunichGermany
  2. 2.Zentrum Mathematik, Bereich M5Technische Universität MünchenGarching bei MünchenGermany
  3. 3.NYU-ECNU Institute of Mathematical Sciences at NYU ShanghaiCourant Institute of Mathematical SciencesNew YorkUSA
  4. 4.CNRS & CEREMADE, Université Paris-DauphinePSL Research UniversityParisFrance

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