The complex elliptic Ginibre ensemble at weak non-Hermiticity: edge spacing distributions

Thomas Bothner*, Alex Little

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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The focus of this paper is on the distribution function of the rightmost eigenvalue for the complex elliptic Ginibre ensemble in the limit of weak non-Hermiticity. We show how the limiting distribution function can be expressed in terms of an integro-differential Painlev ́e-II function and how the same captures the non-trivial transition between Poisson and Airy point process extreme value statistics as the degree of non-Hermiticity decreases. Our most explicit new results concern the tail asymptotics of the limiting distribution function. For the right tail we compute the leading order asymptotics uniformly in the degree of non-Hermiticity, for the left tail we compute it close to Hermiticity.
Original languageEnglish
JournalRandom Matrices: Theory and Applications
Early online date7 May 2024
Publication statusE-pub ahead of print - 7 May 2024


  • Complex elliptic Ginibre ensemble,
  • Fredholm determinants
  • extreme value statistics
  • integro- differential Painlev ́e functions
  • Tracy-Widom and Gumbel distributions
  • Riemann-Hilbert problem
  • nonlinear steepest descent method


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