On a Rosenzweig-Porter-type model

Abstract

We consider a very general Rosenzweig-Porter-type model, $H=H_0+λW$, where $H_0$ is an arbitrary Hermitian matrix and $W$ is a standard Wigner matrix. We precisely trace the localization properties of the eigenvectors and the eigenstate thermalisation hypothesis (ETH) as the coupling constant $λ$ interpolates between the trivial $λ=0$ case and the fully mean field regime of large $λ$. Our results hold uniformly in $H_0$ and $λ$, substantially generalising all previous local laws on deformed Wigner matrices even in the mean field regime. Our proof precisely captures the deterministic approximation to the resolvent which exhibits a strongly inhomogeneous structure. As a byproduct, we conclude the emergence of a mobility edge and study the phenomenon of re-entrant localization.