Closest Spacing of Eigenvalues

Abstract

We study the distribution of the minimum spacing between eigenvalues of a random n by n unitary matrix. The minimum spacing scales as $n^{-4/3}$, not $n^{-2}$ as would be the case for n independent points on the unit circle, illustrating the well known phenomenon that the eigenvalues of random matrices 'repel each other'. We derive the distribution for the rescaled minimum spacing in the limit as $n\to\infty$. To find the minimum spacing, we count the number of eigenvalue pairs closer than $xn^{-4/3}$. We use heuristics to guess that this integer-valued random variable is Poisson, calculate the actual moments of the limiting distribution, and find that the actual moments match those of the guess. The matching moments prove that the heuristic guess is correct, and lead directly to the main result. We prove analogous results for the Gaussian unitary ensemble (GUE) and, with restrictions, a universal class of unitary ensembles (UUE) studied by Deift, Kreicherbauer, McLaughlin, Venakides, and Zhou.