For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the kth largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the kth and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.

We study the joint probability generating function for k occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. This generalizes a result of Tracy and Widom [24], which corresponds to the case k = 1. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large n asymptotics for n × n Hankel determinants with a Laguerre weight possessing several jumps discontinuities near the hard edge. 2(x − y), α > −1, (1.1)where J α stands for the Bessel function of the first kind of order α (see [22, formula 10.2.2] for a definition of J α ).

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We study the eigenvalue correlations of random Hermitian n × n matrices of the form S = M + ǫH, where H is a GUE matrix, ǫ > 0, and M is a positive-definite Hermitian random matrix, independent of H, whose eigenvalue density is a polynomial ensemble. We show that there is a soft-to-hard edge transition in the microscopic behaviour of the eigenvalues of S close to 0 if ǫ tends to 0 together with n → +∞ at a critical speed, depending on the random matrix M . In a double scaling limit, we obtain a new family of limiting eigenvalue correlation kernels. We apply our general results to the cases where (i) M is a Laguerre/Wishart random matrix, (ii) M = G * G with G a product of Ginibre matrices, (iii) M = T * T with T a product of truncations of Haar distributed unitary matrices, and (iv) the eigenvalues of M follow a Muttalib-Borodin biorthogonal ensemble.

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