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 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 \aᵢ\ and a structurally disordered hopping, we found that each eigenstate is delocalised over N^2- sites close in energy |aⱼ-aᵢ| N^1- in agreement with Kravtsov et al, arXiv: 1508. 01714. 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.