$\mathbb{H}^{2|2}$-model and Vertex-Reinforced Jump Process the Regular Trees: Infinite-Order Transition and an Intermediate Phase

Peter Wildemann, Rémy Poudevigne

Published 2023-09-03Version 1

We explore the supercritical phase of the vertex-reinforced jump process (VRJP) and the $\mathbb{H}^{2|2}$-model on rooted regular trees. The VRJP is a random walk, which is more likely to jump to vertices on which it has previously spent a lot of time. The $\mathbb{H}^{2|2}$-model is a supersymmetric lattice spin model, originally introduced as a toy model for the Anderson transition. On infinite rooted regular trees, the VRJP undergoes a recurrence/transience transition controlled by an inverse temperature parameter $\beta > 0$. Approaching the critical point from the transient regime, $\beta \searrow \beta_{\mathrm{c}}$, we show that the expected total time spent at the starting vertex diverges as $\sim \exp(c/\sqrt{\beta - \beta_{\mathrm{c}}})$. Moreover, on large finite trees we show that the VRJP exhibits an additional intermediate regime for parameter values $\beta_{\mathrm{c}} < \beta < \beta_{\mathrm{c}}^{\mathrm{erg}}$. In this regime, despite being transient in infinite volume, the VRJP on finite trees spends an unusually long time at the starting vertex with high probability. We comment on the absence of such an intermediate phase for wired boundary conditions, as a consequence of recent results by Rapenne. We translate these results to the $\mathbb{H}^{2|2}$-model. This provides rigorous evidence for predictions previously made in the physics literature on the Anderson transition. We also demonstrate an approach to study the $\mathbb{H}^{2|2}$-model on trees via a recursive integral equation in polar coordinates for $\mathbb{H}^{2|2}$. Our proofs rely on the application of branching random walk methods to a horospherical marginal of the $\mathbb{H}^{2|2}$-model.