Localization and delocalization for heavy tailed band matrices

TL;DR

This paper investigates the spectral properties of heavy-tailed random band matrices, revealing a phase transition in eigenvalue behavior and eigenvector localization depending on the tail index and bandwidth.

Contribution

It establishes a phase transition in eigenvalue distribution and eigenvector localization for heavy-tailed band matrices, extending previous results to this specific matrix class.

Findings

For <2(1+), largest eigenvalues follow a Poisson process and eigenvectors are localized.

For >2(1+), eigenvalues scale as N^{/2} and eigenvectors are delocalized.

Identifies a critical threshold for the phase transition in spectral behavior.

Abstract

We consider some random band matrices with band-width $N^\mu$ whose entries are independent random variables with distribution tail in $x^{-\alpha}$. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $\alpha\textless{}2(1+\mu^{-1})$, the largest eigenvalues have order $N^{(1+\mu)/\alpha}$, are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked by Soshnikov for full matrices with heavy tailed entries,i.e. when $\alpha\textless{}2$, and by Auffinger, Ben Arous and P{\'e}ch{\'e} when $\alpha\textless{}4$). On the other hand, when $\alpha\textgreater{}2(1+\mu^{-1})$, the largest eigenvalues have order $N^{\mu/2}$ and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their $N$ coordinates.