Random band matrices

TL;DR

This paper surveys recent mathematical results on the spectral properties of random band matrices, focusing on the Anderson transition, delocalization, and universality, with an emphasis on quantum ergodicity.

Contribution

It introduces a method to achieve delocalization and universality in sparse regimes of random band matrices, extending previous approaches to new contexts.

Findings

Application of the Erdős-Schlein-Yau dynamic approach to band matrices

Identification of regimes where delocalization occurs

Connection between quantum ergodicity and spectral universality

Abstract

We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erd{\H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models. We then introduce random band matrices and the problem of their Anderson transition. We finally describe a method to obtain delocalization and universality in some sparse regimes, highlighting the role of quantum unique ergodicity.