Extreme gaps between eigenvalues of random matrices

TL;DR

This paper investigates the behavior of extreme eigenvalue gaps in random matrices, deriving their limiting distributions and comparing these findings with the gaps between zeros of the Riemann zeta function.

Contribution

It provides the joint limiting laws for the smallest and largest eigenvalue gaps in Haar-distributed unitary matrices and Gaussian unitary ensembles, a novel analysis in random matrix theory.

Findings

Smallest gaps follow a specific limiting density proportional to x^{3k-1}e^{-x^3}.

Largest gaps, normalized appropriately, converge to a constant in L^p.

Results are compared with the extreme gaps between zeros of the Riemann zeta function.

Abstract

This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth smallest gap, normalized by a factor $n^{-4/3}$, has a limiting density proportional to $x^{3k-1}e^{-x^3}$. Concerning the largest gaps, normalized by $n/\sqrt{\log n}$, they converge in ${\mathrm{L}}^p$ to a constant for all $p>0$. These results are compared with the extreme gaps between zeros of the Riemann zeta function.