Non-Ergodic Delocalization in the Rosenzweig-Porter Model

We consider the Rosenzweig-Porter model $H=V+\sqrt{T}\mathrm{\Phi }$, where $V$ is a $N\times N$ diagonal matrix, $\mathrm{\Phi }$ 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 $H$.