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

TL;DR

This paper derives the joint distribution of the first two eigenvalues at the soft edge of unitary ensembles using Painlevé II transcendent, providing new analytical formulas and numerical methods for eigenvalue spacing.

Contribution

It introduces a novel characterization of the joint eigenvalue distribution at the soft edge via Painlevé II and extends existing results from the hard edge through edge transition techniques.

Findings

Joint distribution expressed via Painlevé II transcendent

Explicit formulas for eigenvalue spacing distribution

Numerical methods for accurate eigenvalue spacing computation

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 Painlev\'e 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 Painlev\'e II that arises is a double shifted B\"acklund transformation of the Hasting-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 transitions to existing results for the joint distribution of the first and second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under $a \mapsto a+1$ of quantities specifying the latter are obtained. A Fredholm determinant type characterisation is used to provide accurate numerics for the distribution of the spacing between the two largest eigenvalues.