THE VERTEX REINFORCED JUMP PROCESS AND A RANDOM SCHRODINGER OPERATOR ON FINITE GRAPHS

• 成果类型: Article
• 署名作者: Sabot, Christophe; Tarres, Pierre; Zeng, Xiaolin
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); New York University; NYU Shanghai; Tel Aviv University; Universite PSL; Universite Paris-Dauphine; New York University; NYU Shanghai
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1155
• 发表日期: 2017
• 页码: 3967-3986
• 关键词: reversible markov-chains; bayesian-analysis; sigma-model

摘要

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 Schrodinger 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.