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
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.
Ann. Probab., Volume 45, Number 6A (2017), 3967-3986.
Received: January 2016
Revised: September 2016
First available in Project Euclid: 27 November 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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