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The Vertex Reinforced Jump Process and a random Schrödinger operator on finite graphs

Sabot, Christophe; Tarrès, Pierre; Zeng, Xiaolin (2017), The Vertex Reinforced Jump Process and a random Schrödinger operator on finite graphs, Annals of Probability, 45, 6A, p. 3967 - 3986. 10.1214/16-AOP1155

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Type
Article accepté pour publication ou publié
Date
2017
Journal name
Annals of Probability
Volume
45
Number
6A
Publisher
Institute of Mathematical Statistics
Pages
3967 - 3986
Publication identifier
10.1214/16-AOP1155
Metadata
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Author(s)
Sabot, Christophe
Institut Camille Jordan [Villeurbanne] [ICJ]
Tarrès, Pierre
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Zeng, Xiaolin cc
Institut Camille Jordan [Villeurbanne] [ICJ]
Abstract (EN)
We introduce a new exponential family of probability distributions, which can be viewed as a multivariate generalization of the inverse Gaussian distribution. Considered as the potential of a random Schrödinger operator, this exponential family is related to the random field that gives the mixing measure of the Vertex Reinforced Jump Process (VRJP), and hence to the mixing measure of the Edge Reinforced Random Walk (ERRW), the so-called magic formula. In particular, it yields by direct computation the value of the normalizing constants of these mixing measures, which solves a question raised by Diaconis. The results of this paper are instrumental in [Sabot and Zeng (2015)], where several properties of the VRJP and the ERRW are proved, in particular a functional central limit theorem in transient regimes, and recurrence of the 2-dimensional ERRW.
Subjects / Keywords
random Schrödinger operator; Self-interacting random walks; supersymmetric hyperbolic nonlinear sigma model; Vertex-reinforced jump process

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