A continuous random operator associated with the Vertex Reinforced Jump Process on the circle and the real line

TL;DR

This paper studies the scaling-limit of a random potential linked to the Vertex Reinforced Jump Process, constructing a continuous random Schrödinger operator on the circle and real line, and deriving new probabilistic identities.

Contribution

It introduces a continuous-space version of the random Schrödinger operator associated with VRJP and provides new proofs and identities involving Brownian motion and exponential functionals.

Findings

Explicit form of the integrated density of states for the operator on R

New proof of Matsumoto-Yor properties for geometric Brownian motion

Generalized identities involving exponential functionals of Brownian motion

Abstract

In this paper, we focus on the scaling-limit of the random potential $\beta$ associated with the Vertex Reinforced Jump Process (VRJP) on one-dimensional graphs. Moreover, we give a few applications of this scaling-limit. By considering a relevant scaling of $\beta$, we contruct a continuous-space version of the random Schr{\"o}dinger operator $H_\beta$ which is associated with the VRJP on circles and on R. We also compute the integrated density of states of this operator on R which has a remarkably simple form. Moreover, by means of the same scaling, we obtain a new proof of the Matsumoto-Yor properties concerning the geometric Brownian motion which were proved in [MY01]. This new proof is based on some fundamental properties of the random potential $\beta$. We use also the scaling-limit of $\beta$ in order to prove new identities in law involving exponential functionals of the Brownian motion which generalize the Dufresne identity.