Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlevé II system

Abstract

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.

57 pages, 12 figures, references updated. The constant terms in the large gap asymptotics are evaluated explicitly in this version

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• Randomly incomplete spectra and intermediate statistics
• Monodromy dependence and connection formulae for isomonodromic tau functions
• Large deformations of the Tracy-Widom distribution I. Non-oscillatory asymptotics
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Cited by in corpus (6)

• Large gap asymptotics for Airy kernel determinants with discontinuities
• Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
• Gap probability of the circular unitary ensemble with a Fisher-Hartwig singularity and the coupled Painlevé V system
• Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
• Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System
• Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlevé XXXIV equations