Higher Order Analogues of Tracy-Widom Distributions via the Lax Method

Abstract

We study the distribution of the largest eigenvalue in formal Hermitian one-matrix models at multicriticality, where 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 non-linear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990's. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painleve XXXIV hierarchy. These are the higher order analogues of the Tracy-Widom distribution which has k=1. Using known Backlund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painleve II hierarchy.

arXiv comment24 pages. Improved discussion of Backlund transformations, in addition to other minor improvements in text. Typos corrected. Matches published version

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• Tracy-Widom distribution as instanton sum of 2D IIA superstrings
• Instantons and Extreme Value Statistics of Random Matrices
• Non-probabilistic fermionic limit shapes
• Random matrices in non-confining potentials
• Classical group matrix models and universal criticality
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• Universal Cusp Scaling in Random Partitions