Discrete Tracy-Widom Operators

TL;DR

This paper investigates discrete Tracy-Widom operators in random matrix theory, providing conditions under which these operators are squares of Hankel matrices, with applications to Bessel kernels and Mathieu equations.

Contribution

It establishes sufficient conditions for discrete integrable operators to be expressed as squares of Hankel matrices, expanding understanding of their structure and applications.

Findings

Discrete Tracy-Widom operators can be represented as squares of Hankel matrices under certain conditions.

Examples include discrete Bessel kernels and kernels from Mathieu equations.

The results connect integrable operators with classical special functions.

Abstract

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. This paper considers discrete Tracy-Widom operators, and gives sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.