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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 arandom walk, which is more likely to jump to vertices on which it haspreviously spent a lot of time. The $\mathbb{H}^{2|2}$-model is asupersymmetric lattice spin model, originally introduced as a toy model for theAnderson transition. On infinite rooted regular trees, the VRJP undergoes arecurrence/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 thestarting vertex diverges as $\sim \exp(c/\sqrt{\beta - \beta_{\mathrm{c}}})$.Moreover, on large finite trees we show that the VRJP exhibits an additionalintermediate regime for parameter values $\beta_{\mathrm{c}} < \beta <\beta_{\mathrm{c}}^{\mathrm{erg}}$. In this regime, despite being transient ininfinite volume, the VRJP on finite trees spends an unusually long time at thestarting vertex with high probability. We comment on the absence of such anintermediate phase for wired boundary conditions, as a consequence of recentresults by Rapenne. We translate these results to the $\mathbb{H}^{2|2}$-model.This provides rigorous evidence for predictions previously made in the physicsliterature on the Anderson transition. We also demonstrate an approach to studythe $\mathbb{H}^{2|2}$-model on trees via a recursive integral equation inpolar coordinates for $\mathbb{H}^{2|2}$. Our proofs rely on the application ofbranching random walk methods to a horospherical marginal of the$\mathbb{H}^{2|2}$-model.

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