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For a wide class of Hermitian random matrices, the limit distribution of theeigenvalues close to the largest one is governed by the Airy point process. Insuch ensembles, the limit distribution of the k-th largest eigenvalue is givenin terms of the Airy kernel Fredholm determinant or in terms of Tracy-Widomformulas involving solutions of the Painlev\'e II equation. Limit distributionsfor quantities involving two or more near-extreme eigenvalues, such as the gapbetween the k-th and the \ell-th largest eigenvalue or the sum of the k largesteigenvalues, can be expressed in terms of Fredholm determinants of an Airykernel with several discontinuities. We establish simple Tracy-Widom typeexpressions for these Fredholm determinants, which involve solutions to systemsof coupled Painlev\'e II equations, and we investigate the asymptotic behaviorof these solutions.

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