The generating function for the Airy point process and a system of coupled Painlev\'e II equations

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 k-th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy-Widom formulas involving solutions of the Painlev\'e II equation. Limit distributions for quantities involving two or more near-extreme eigenvalues, such as the gap between the k-th and the \ell-th 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 Painlev\'e II equations, and we investigate the asymptotic behavior of these solutions.