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We prove the vertex-reinforced jump process (VRJP) is recurrent in twodimensions for any translation invariant finite range initial rates. Our proofhas two main ingredients. The first is a direct connection between the VRJP andsigma models whose target space is a hyperbolic space $\mathbb{H}^n$ or itssupersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues ofwell-known relations between the Gaussian free field and the local times ofsimple random walk. The second ingredient is a Mermin--Wagner theorem for thesesigma models. This result is of intrinsic interest for the sigma models andalso implies our main theorem on the VRJP. Surprisingly, our Mermin--Wagnertheorem applies even though the symmetry groups of $\mathbb{H}^n$ and$\mathbb{H}^{2|2}$ are non-amenable.

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