Smallest Gaps Between Eigenvalues of Random Matrices With Complex Ginibre, Wishart and Universal Unitary Ensembles

Abstract

In this paper we study the limiting distribution of the $k$ smallest gaps between eigenvalues of three kinds of random matrices -- the Ginibre ensemble, the Wishart ensemble and the universal unitary ensemble. All of them follow a Poissonian ansatz. More precisely, for the Ginibre ensemble we have a global result in which the $k$-th smallest gap has typical length $n^{-3/4}$ with density $x^{4k-1}{e}^{-x^4}$ after normalization. For the Wishart and the universal unitary ensemble, it has typical length $n^{-4/3}$ and has density $x^{3k-1}{e}^{-x^3}$ after normalization.

31 pages, 1 figure