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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
Article
  • 97 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

Primary 60K35 Secondary 81T60 

Notes

Acknowledgements

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

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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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