Similarity between a many-body quantum avalanche model and the ultrametric random matrix model

In the field of ergodicity-breaking phases, it has been recognized that quantum avalanches can destabilize many-body localization at a wide range of disorder strengths. This has in particular been demonstrated by the numerical study of a toy model, sometimes simply called the "avalanche model" or the "quantum sun model" [Phys. Rev. Lett. 129, 060602 (2022)], which consists of an ergodic seed coupled to a perfectly localized material. In this paper, we connect this toy model to a well-studied model in random matrix theory, the ultrametric ensemble. We conjecture that the models share the following features. 1) The location of the critical point may be predicted sharply by analytics. 2) On the localized site, both models exhibit Fock space localization. 3) There is a manifold of critical points. On the critical manifold, the eigenvectors exhibit nontrivial multifractal behaviour that can be tuned by moving on the manifold. 4) The spectral statistics at criticality is intermediate between Poisson statistics and random matrix statistics, also tunable on the critical manifold. We confirm numerically these properties.