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 Hⁿ or its supersymmetric counterpart 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 Hⁿ and H^2|2 are nonamenable.