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

Merkl, Franz; Rolles, Silke; Tarres, Pierre (2018), Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model, Probability Theory and Related Fields, 173, p. 1349–1387. 10.1007/s00440-018-0855-8

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Type
Article accepté pour publication ou publié
Date
2018
Journal name
Probability Theory and Related Fields
Number
173
Publisher
Springer
Pages
1349–1387
Publication identifier
Metadata
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Author(s)
Merkl, Franz
Mathematical Institute
Rolles, Silke
Zentrum Mathematik [Munchen] [TUM]
Tarres, Pierre
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Courant Institute of Mathematical Sciences [New York] [CIMS]
Abstract (EN)
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.
Subjects / Keywords
Vertex-reinforced jump process; Self-interacting random walks; Supersymmetric hyperbolic nonlinear sigma model