Dynkin isomorphism and Mermin--Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

TL;DR

This paper establishes the recurrence of the two-dimensional vertex-reinforced jump process by connecting it to hyperbolic sigma models and proving a Mermin--Wagner theorem for these models, despite their non-amenable symmetry groups.

Contribution

It introduces a novel link between VRJP and hyperbolic sigma models, and proves a Mermin--Wagner theorem for non-amenable symmetry groups, leading to new recurrence results.

Findings

VRJP is recurrent in two dimensions for finite range initial rates

Established a Mermin--Wagner theorem for hyperbolic sigma models with non-amenable groups

Connected VRJP behavior to hyperbolic sigma models and their symmetries

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 non-amenable.