-We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter N × N random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of onsite random energies {ai} and a structurally disordered hopping, we found that each eigenstate is delocalised over N 2−γ sites close in energy |aj − ai| ≤ N 1−γ in agreement with Kravtsov et al. (New. J. Phys., 17 (2015) 122002) . Our other main result, obtained combining a recurrence relation for the resolvent matrix with insights from Dyson's Brownian motion, is to show that the properties of the non-ergodic delocalised phase can be probed studying the statistics of the local resolvent in a non-standard scaling limit.The theoretical study of the non-equilibrium dynamics of isolated quantum systems has attracted considerable interest in recent years, partly due to advances in experiments on trapped ultra-cold atomic gases [1]. One of the most fundamental questions that arose is about the applicability of statistical mechanics to quantum systems in presence of interactions and disorder, and the related Many-body localisation (MBL) transition [2]. A system is in a MBL phase if taking interactions into account the many-body eigenstates are localised in Fock space. The Fock space can be seen as a lattice with connectivity determined by two-body interactions. Its structure is that of a very high dimensional lattice where loops are scarce, therefore reminiscent of the Bethe lattice and random regular graphs (RRG). Starting from the pioneering work [3], Anderson localization on such lattices has been considered by many as a simplified case to study questions related to the MBL transition. It attracted a lot of attention recently [4-6] because it could provide a test ground to analyse the delocalised non-ergodic or "bad metal" regime, which was predicted as an intermediate phase separating the fully delocalised and the MBL phases [2,3]. In (a) davide.facoetti@kcl.ac.uk this regime, eigenstates would be delocalised over a large number of configurations, but which only cover a very tiny fraction, vanishing for large system size, of the entire Fock space. Although the existence of the MBL transition is now well established (at least for one dimensional systems) [7], the understanding of the delocalised non-ergodic phase is far from being completed. Some numerical results seem to indicate its presence in many-body systems [8,9] whereas its existence on Bethe lattices is under intense scrutiny and debated [4,6,[10][11][12]. It is not clear at this stage whether the sub-diffusive behaviour found before the MBL transition [13][14][15][16][17] is somehow related to it.Given this state of the art, it is therefore useful to study simpler models that could provide a playground to explore its nature and sharpen the questions about it. With this aim, the authors of Ref. [18] proposed a random matrix model, the generalised Rosenzweig-Porter (GRP) model,...

From the dynamics of a broad class of classical mean-field glass models one may obtain a quantum model with finite zero-temperature entropy, a quantum transition at zero temperature, and a timereparametrization (quasi-)invariance in the dynamical equations for correlations. The low eigenvalue spectrum of the resulting quantum model is directly related to the structure and exploration of metastable states in the landscape of the original classical glass model. This mapping reveals deep connections between classical glasses and the properties of SYK-like models.