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 $β_n^{3/4}\bigl(\sqrt n\,M_n(B)-β_n^{1/4}\bigr)$ converges in distribution to a Gumbel random variable, and we determine $β_n$ explicitly.

Disclosure

“nts P.M. was supported by the Swedish Foundation for International Cooperation in Research and Higher Education (PD2023-9315). Statement on the use of artificial intelligence In the course of the research presented here I have been using AI tools extensively, most significantly ChatGPT Pro 5.5 and ChatGPT 5.6 Sol Ultra, for ideation, technical help, editing and checking the proofs, and general editing and proofreading of the manuscript, at the level of a leading co-author. All mist”

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2026-08-22T03:15:10.498058Z

Structural counts