King's College London

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 {a_i} and a structurally disordered hopping, we found that each eigenstate is delocalised over N^2-γ sites close in energy |a_j - a_i| ≤ 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.