The vertex-reinforced jump process (VRJP) is a linearly reinforced random walk in continuous time. Reinforcing means that the VRJP prefers to jump to vertices it has previously visited; the strength of the reinforcement is a parameter of the model.

Sabot and Tarres have shown that the VRJP is related to a model known as the H22 supersymmetric hyperbolic spin model, which originated in the study of random band matrices. By making use of results for the H22 model they proved the VRJP is recurrent for sufficiently strong reinforcement. I will present a new and direct connection between the VRJP and hyperbolic spin models (both supersymmetric and classical), and show how this connection can be used to prove that the VRJP is recurrent in two dimensions for all reinforcement strengths.

This talk is about joint work with R. Bauerschmidt and A. Swann.