The spectral edge of some random band matrices

TL;DR

This paper investigates the eigenvalue distributions at the spectral edges of random Hermitian band matrices, revealing a phase transition in the limiting distribution depending on the band width.

Contribution

It establishes the precise band width threshold (W >> N^{5/6}) for the eigenvalues near the spectral edges to follow the Airy point process, highlighting a new phase transition.

Findings

Eigenvalues near spectral edges follow Airy point process when band width W >> N^{5/6}.

Different limiting distributions emerge when W is below the threshold.

The study characterizes the spectral edge behavior for a class of random band matrices.

Abstract

We study the asymptotic distribution of the eigenvalues of random Hermitian periodic band matrices, focusing on the spectral edges. The eigenvalues close to the edges converge in distribution to the Airy point process if (and only if) the band is sufficiently wide (W >> N^{5/6}.) Otherwise, a different limiting distribution appears.