Non-ergodic delocalization in the Rosenzweig–Porter model

We consider the Rosenzweig–Porter model H=V+TΦ, where V is a N× N diagonal matrix, Φ 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.