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We study the statistics of the local resolvent and non-ergodic properties ofeigenvectors for a generalised Rosenzweig-Porter $N\times N$ random matrixmodel, 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 isdelocalised over $N^{2-\gamma}$ sites close in energy $|a_j-a_i|\leqN^{1-\gamma}$ in agreement with Kravtsov \emph{et al}, arXiv:1508.01714. Ourother main result, obtained combining a recurrence relation for the resolventmatrix with insights from Dyson's Brownian motion, is to show that theproperties of the non-ergodic delocalised phase can be probed studying thestatistics of the local resolvent in a non-standard scaling limit.

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