## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 6A (2017), 3967-3986.

### The Vertex Reinforced Jump Process and a random Schrödinger operator on finite graphs

Christophe Sabot, Pierre Tarrès, and Xiaolin Zeng

#### 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 Schrödinger 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 and 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.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 6A (2017), 3967-3986.

**Dates**

Received: January 2016

Revised: September 2016

First available in Project Euclid: 27 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1511773669

**Digital Object Identifier**

doi:10.1214/16-AOP1155

**Mathematical Reviews number (MathSciNet)**

MR3729620

**Zentralblatt MATH identifier**

06838112

**Subjects**

Primary: 60K37: Processes in random environments 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Secondary: 81T25: Quantum field theory on lattices 81T60: Supersymmetric field theories

**Keywords**

Vertex-reinforced jump process self-interacting random walks random Schrödinger operator supersymmetric hyperbolic nonlinear sigma model

#### Citation

Sabot, Christophe; Tarrès, Pierre; Zeng, Xiaolin. The Vertex Reinforced Jump Process and a random Schrödinger operator on finite graphs. Ann. Probab. 45 (2017), no. 6A, 3967--3986. doi:10.1214/16-AOP1155. https://projecteuclid.org/euclid.aop/1511773669