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 $N\times N$ 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 $N^{2-\gamma}$ sites close in energy $|a_j-a_i|\leq N^{1-\gamma}$ in agreement with Kravtsov \emph{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.