Extremal eigenvalue fluctuations in the GUE minor process and the law of fractional logarithm

TL;DR

This paper studies the fluctuations of the largest eigenvalues in the GUE minor process, establishing a law of fractional logarithm that describes their almost sure asymptotic behavior.

Contribution

It introduces a law of iterated logarithm analogue for the GUE minor process eigenvalues, revealing their almost sure fluctuation bounds.

Findings

Limsup of normalized eigenvalues converges to a constant almost surely.

Provides almost sure bounds for scaled liminf of eigenvalues.

Establishes a fractional logarithm law for eigenvalue fluctuations.

Abstract

We consider the GUE minor process, where a sequence of GUE matrices is drawn from the corner of a doubly infinite array of i.i.d. standard normal variables subject to the symmetry constraint. From each matrix, we take its largest eigenvalue, appropriately rescaled to converge to the standard Tracy-Widom distribution. We show the analogue of the law of iterated logarithm for this sequence, i.e. we divide the normalized n-th eigenvalue by a logarithmic factor and show the limsup of this sequence is a constant almost surely. We also give almost sure bounds for the appropriately scaled liminf.