The vertex-reinforced jump process (VJRP) is a random walk that prefers to jump to vertices visited in the past. Hyperbolic sigma models are
spin models where the spins take values in a hyperbolic space. I will explain a relation between the VRJP and hyperbolic sigma models which
parallels that between the simple random walk and the Gaussian free field. I will further show a Mermin-Wagner Theorem for hyperbolic sigma
models which implies that the VRJP is recurrent in two dimensions.

This is joint work with Tyler Helmuth and Andrew Swan.