Largest bulk gap of the complex Ginibre ensemble

Abstract

Let $M_n(B)$ be the largest distance from an eigenvalue of an $n\times n$ complex Ginibre matrix, with entries of variance $1/n$, lying in a fixed bulk set $B$ compactly contained in the unit disk and of planar area $|B|$, to its nearest other eigenvalue. Lopatto and Otto proved that $ \sqrt{n} M_n(B)/(4\log n)^{1/4} \to 1$ in probability. Here we prove that $\beta_n^{3/4}\bigl(\sqrt n\,M_n(B)-\beta_n^{1/4}\bigr)$ converges in distribution to a Gumbel random variable, and we determine $\beta_n$ explicitly.