Higher order analogues of Tracy–Widom distributions via the Lax method

Abstract

We study the distribution of the largest eigenvalue in Hermitian one-matrix models when the spectral density acquires an extra number of k − 1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy nonlinear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990s. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painlevé XXXIV hierarchy. These are the higher order analogues of the Tracy–Widom distribution. Using known Bäcklund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painlevé II hierarchy.