Non-Ergodic Delocalization in the Rosenzweig-Porter Model

We consider the Rosenzweig-Porter model $H = V + sqrt{T}, Phi$, where $V$ is a $N times N$ diagonal matrix, $Phi$ is drawn from the $N times N$ Gaussian Orthogonal Ensemble, and $N^{-1} ll T ll 1$. 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 $H$.