Reinforced Random Walk and a Supersymmetric Spin System on the Tree
Motivated by predictions about the Anderson transition, we study two distinct but related models on regular tree graphs: The vertex-reinforced jump process (VRJP), a random walk preferring to jump to previously visited sites, and the H^{2|2}-model, a lattice spin system whose spins take values in a supersymmetric extension of the hyperbolic plane. Both models undergo a phase transition, and our work provides detailed information about the supercritical phase up to the critical point: We show that their order parameter has an essential singularity as one approaches the critical point, in contrast to algebraic divergences typically expected in statistical mechanics. Moreover, we locate an additional multifractal intermediate phase on large finite trees. This talk is based on arxiv:2309.01221 and is joint work with Rémy Poudevigne.