Largest bulk gap of the complex Ginibre ensemble

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