Convergence of vertex-reinforced jump processes to an extension of the supersymmetric hyperbolic nonlinear sigma model

Abstract

In this paper, we define an extension of the supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer. We show that it arises as a weak joint limit of a time-changed version introduced by Sabot and Tarrès of the vertex-reinforced jump process. It describes the asymptotics of rescaled crossing numbers, rescaled fluctuations of local times, asymptotic local times on a logarithmic scale, endpoints of paths, and last exit trees. 1

Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

“…The random variables u encode the asymptotics of local times for a time changed vertex reinforced jump process while the random variables s describe the corresponding fluctuations. For details see [MRT16].…”

Explicit formula for the density of local times of Markov jump processes

“…We first show the following Proposition 2.1. Its proof relies on an argument similar to the proof of Theorem 1.6 in [7]; the technique for determining the cardinality of the set of paths with given last exit tree are from Lemma 6 by Keane and Rolles in [3].…”

“…Explicit formulas for the joint density of local times of continous-time Markov Chains were already proposed, see for instance [6,1]. Merkl, Rolles and Tarrès proposed in [7] a formula for the joint density of the oriented edge crossings, local times and last-exit tree for the Vertex-Reinforced Jump Process on a general graph, whose counterpart in the context of continuous-time Markov Chains is stated in Proposition 2.1 below.…”

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

“…Surprisingly, although the model was originally designed to describe condensed matter systems, there is a close connection with vertex-reinforced jump processes as was observed by Sabot and Tarrès [ST15]. All spin variables of the H 2|2 -model written in horospherical coordinates have a probabilistic interpretation in terms of the vertex-reinforced jump process as is worked out in [MRT19].…”