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

• Christophe Sabot ^[3] ; Pierre Tarrès ^[1] ; Xiaolin Zeng ^[2]
  1. [1] New York University

     New York University

     Estados Unidos

     
  2. [2] Tel Aviv University

     Tel Aviv University

     Israel

     
  3. [3] Université Lyon (France)

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 6, 1, 2017, págs. 3967-3986
• Idioma: inglés
• DOI: 10.1214/16-AOP1155
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• Resumen

  • 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.