From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective

We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter N x N random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of on-site random energies {a(i)} and a structurally disordered hopping, we found that each eigenstate is delocalised over N2-gamma sites close in energy vertical bar a(j) - a(i)vertical bar = N1-gamma 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. Copyright (C) EPLA, 2016