The Vertex Reinforced Jump Process and a Random Schr\"odinger operator on finite graphs
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 Schr\"odinger 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-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.
Keywords
Cite
@article{arxiv.1507.04660,
title = {The Vertex Reinforced Jump Process and a Random Schr\"odinger operator on finite graphs},
author = {Christophe Sabot and Pierre Tarrès and Xiaolin Zeng},
journal= {arXiv preprint arXiv:1507.04660},
year = {2016}
}
Comments
15 pages