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November 2017 The Vertex Reinforced Jump Process and a random Schrödinger operator on finite graphs
Christophe Sabot, Pierre Tarrès, Xiaolin Zeng
Ann. Probab. 45(6A): 3967-3986 (November 2017). DOI: 10.1214/16-AOP1155

Abstract

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.

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Christophe Sabot. Pierre Tarrès. Xiaolin Zeng. "The Vertex Reinforced Jump Process and a random Schrödinger operator on finite graphs." Ann. Probab. 45 (6A) 3967 - 3986, November 2017. https://doi.org/10.1214/16-AOP1155

Information

Received: 1 January 2016; Revised: 1 September 2016; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06838112
MathSciNet: MR3729620
Digital Object Identifier: 10.1214/16-AOP1155

Subjects:
Primary: 60K35 , 60K37 , 82B44
Secondary: 81T25 , 81T60

Keywords: random Schrödinger operator , Self-interacting random walks , supersymmetric hyperbolic nonlinear sigma model , Vertex-reinforced jump process

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 6A • November 2017
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