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

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.