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

TL;DR

This paper introduces higher order Tracy-Widom distributions for multicritical Hermitian matrix models, expressing them via orthogonal polynomials, Lax pairs, and Painleve hierarchies, extending classical results.

Contribution

It provides a new approach to higher order Tracy-Widom distributions using orthogonal polynomials and Lax pairs, connecting to Painleve hierarchies and simplifying previous Fredholm determinant methods.

Findings

Distributions expressed through orthogonal polynomial norms

Distribution satisfies non-linear recurrence relations forming a Lax pair

Explicit formulas derived using Painleve XXXIV hierarchy

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.