Non-ergodic delocalization in the Rosenzweig–Porter model

We consider the Rosenzweig–Porter model H = V + √(T) , where V is a N × N diagonal matrix, is drawn from the N × N Gaussian Orthogonal Ensemble, and N^-1≪ T ≪ 1 . We prove that the eigenfunctions of H are typically supported in a set of approximately NT sites, thereby confirming the existence of a previously conjectured non-ergodic delocalized phase. Our proof is based on martingale estimates along the characteristic curves of the stochastic advection equation satisfied by the local resolvent of the Brownian motion representation of H .