Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles

Abstract

The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painlevé II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterized.The particular solution of Painlevé II that arises is a double shifted Bäcklund transformation of the Hastings-McLeod solution, which applies in the case of the distribution of the largest eigenvalue at the soft edge. Our deductions are made by employing the hard-to-soft edge transition, involving the limit as the repulsion strength at the hard edge a → ∞, to existing results ..