Localization of One-Dimensional Random Band Matrices

TL;DR

This paper proves that eigenvectors of certain one-dimensional random band matrices are localized when the bandwidth is small relative to the matrix size, confirming a key aspect of the localization-delocalization transition conjecture.

Contribution

It establishes the exponential localization of eigenvectors for random band matrices with bandwidth W when W^2 is much less than n, completing the understanding of the transition.

Findings

Eigenvectors are localized with exponential decay at scale W^2

Confirms the localization-delocalization transition conjecture

Complements previous delocalization results by Yau, Yin, Erdős, and Riabov

Abstract

We consider a general class of $n\times n$ random band matrices with bandwidth $W$. When $W^2\ll n$, we prove that with high probability the eigenvectors of such matrices are localized and decay exponentially at the sharp scale $W^2$. Combined with the delocalization results of Yau and Yin [arXiv:2501.01718] and of Erd\H{o}s and Riabov [arXiv:2506.06441], this establishes the conjectured localization-delocalization transition for a large class of random band matrices.