The *-Vertex-Reinforced Jump Process

TL;DR

This paper explores a non-reversible generalization of the Vertex-Reinforced Jump Process, introducing new representations and phenomena, including a mixture of Markov processes and a connection to a random Schrödinger operator.

Contribution

It introduces the *-VRJP, a non-reversible extension of VRJP, and provides novel representations involving mixtures of Markov processes and a random Schrödinger operator.

Findings

Partial exchangeability after randomization of initial local time

Representation of *-VRJP as a mixture of Yaglom reversible Markov processes

Connection between *-VRJP and a random Schrödinger operator

Abstract

We investigate the non-reversible generalization of the Vertex-Reinforced Jump Process (VRJP), called the *-Vertex-Reinforced Jump Process (*-VRJP) and introduced by Bacallado, Sabot and Tarr\`es (2020). It can be seen as the continuous-time counterpart to the *-Edge-Reinforced Random Walk (*-ERRW), see Bacallado (2011) and Bacallado, Sabot and Tarr\`es (2020), which is itself a non-reversible, and in fact Yaglom reversible, generalization of the original ERRW introduced by Coppersmith and Diaconis (1986). In contrast to the classical VRJP, the *-VRJP is not exchangeable after time-change, which leads to several difficulties and new phenomena. Firstly, we show that with some appropriate randomization of the initial local time, it becomes partially exchangeable after time-change. We provide a representation of the "randomized" *-VRJP as a mixture of Yaglom reversible Markov jump processes with an explicit mixing measure, and we prove that the non-randomized *-VRJP can be written as a mixture of conditioned Markov processes. Secondly, we give a representation of the *-VRJP in terms of a random Schr\"odinger operator. The corresponding representation for the classical VRJP has proved to be very useful in the understanding of its asymptotic behavior. The construction is based on several new and rather remarkable identities between integrals on the space of *-symmetric and *-antisymmetric functions on vertices. We give a description of the randomized *-VRJP in terms of that random Schr\"odinger operator, which allows us to prove the representation of the randomized *-VRJP as a mixture of Markov jump processes in a different and more analytic manner. Similarly as for the VRJP, we think that the representation by a random Schr\"odinger operator and the associated identities are key-features of the *-VRJP.