Communications in Mathematical Physics

, Volume 365, Issue 2, pp 515–567 | Cite as

Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System

  • Shuai-Xia Xu
  • Dan DaiEmail author


We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy–Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlevé II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Franco-Chinois de l’Energie NucléaireSun Yat-sen UniversityGuangzhouChina
  2. 2.Department of MathematicsCity University of Hong KongKowloonHong Kong

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