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September 2019 Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process
Roland Bauerschmidt, Tyler Helmuth, Andrew Swan
Ann. Probab. 47(5): 3375-3396 (September 2019). DOI: 10.1214/19-AOP1343

Abstract

We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite-range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $\mathbb{H}^{n}$ or its supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin–Wagner theorem for these sigma models. This result is of intrinsic interest for the sigma models and also implies our main theorem on the VRJP. Surprisingly, our Mermin–Wagner theorem applies even though the symmetry groups of $\mathbb{H}^{n}$ and $\mathbb{H}^{2|2}$ are nonamenable.

Citation

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Roland Bauerschmidt. Tyler Helmuth. Andrew Swan. "Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process." Ann. Probab. 47 (5) 3375 - 3396, September 2019. https://doi.org/10.1214/19-AOP1343

Information

Received: 1 March 2018; Revised: 1 October 2018; Published: September 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07145320
MathSciNet: MR4021254
Digital Object Identifier: 10.1214/19-AOP1343

Subjects:
Primary: 60G60, 82B20

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 5 • September 2019
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