Non-Ergodic Delocalization in the Rosenzweig-Porter Model

TL;DR

This paper proves the existence of a non-ergodic delocalized phase in the Rosenzweig-Porter model, showing eigenfunctions are supported on about NT sites, confirming a long-standing conjecture.

Contribution

It provides a rigorous proof of non-ergodic delocalization in the Rosenzweig-Porter model using martingale estimates and stochastic advection equations.

Findings

Eigenfunctions are supported on approximately NT sites.

Confirmed the existence of a non-ergodic delocalized phase.

Established a new method using martingale estimates for analysis.

Abstract

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 $H$.