Spacings in Orthogonal and Symplectic Random Matrix Ensembles

TL;DR

This paper proves that the eigenvalue spacing distribution in orthogonal and symplectic invariant random matrix ensembles converges uniformly to a universal limit as matrix size increases.

Contribution

It establishes the uniform convergence of the empirical eigenvalue spacing distribution for orthogonal and symplectic ensembles, extending universality results.

Findings

Expected Kolmogorov distance converges to zero

Empirical spacing distribution converges uniformly

Results hold as matrix size tends to infinity

Abstract

We consider the universality of the nearest neighbour eigenvalue spacing distribution in invariant random matrix ensembles. Focussing on orthogonal and symplectic invariant ensembles, we show that the empirical spacing distribution converges in a uniform way. More precisely, the main result states that the expected Kolmogorov distance of the empirical spacing distribution from its universal limit converges to zero as the matrix size tends to infinity.