How Vertex reinforced jump process arises naturally

We prove that the only nearest neighbor jump process with local dependence on the occupation times satisfying the partial exchangeability property is the vertex reinforced jump process, under some technical conditions. This result gives a counterpart to the characterization of edge reinforced random walk given by Rolles.Comment: 14 pages, 3 figures, version

The Vertex Reinforced Jump Process and a Random Schr\"odinger operator on finite graphs

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.Comment: 15 page

Speed of Vertex reinforced jump process on Galton-Watson trees

International audienceWe give an alternative proof of the fact that the vertex reinforced jump process on Galton-Watson tree has a phase transition between recurrence and transience as a function of c, the initial local time, see [3]. Further, applying techniques in [1], we show a phase transition between positive speed and null speed for the associated discrete time process in the transient regime