Large gap asymptotics for Airy kernel determinants with discontinuities

Abstract

We obtain large gap asymptotics for Airy kernel Fredholm determinants with any number $m$ of discontinuities. These $m$-point determinants are generating functions for the Airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles. Our main result is that the $m$-point determinants can be expressed asymptotically as the product of $m$ $1$-point determinants, multiplied by an explicit constant pre-factor which can be interpreted in terms of the covariance of the counting function of the process.

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• Large gap asymptotics for the generating function of the sine point process
• Exponential moments for disk counting statistics of random normal matrices in the critical regime
• Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation
• The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Large deviation bounds for the Airy point process
• Global rigidity and exponential moments for soft and hard edge point processes