The Instability of Painlev\'e Equations in Recovering Largest Eigenvalue Distributions of GUE, LUE, JUE and an Attempt of Solution to It

Abstract

The distribution of the largest eigenvalue for the three classical unitary ensembles -- GUE, LUE, and JUE -- admits two complementary exact descriptions: (i) as Fredholm determinants of their orthogonal polynomial correlation kernels and (ii) as isomonodromic $\tau$-functions governed by Painlev\'e equations. For finite $n$, the associated Jimbo-Miwa-Okamoto $\sigma$-forms are $\PIV$ (GUE), $\mathrm{PV}$ (LUE), and $\PVI$ (JUE); under soft- or hard-edge scalings these degenerate to $\PII$ or $\PIIIp$ descriptions of the Tracy-Widom and hard-edge laws \cite{tracy1994level,forrester2003painleve,deift1999orthogonal}. It is well known among random matrix theorists (for example Folkmar Bornemann) that the Fredholm determinant is a more numerically stable and accurate way to compute the CDF of the largest eigenvalue for GUE, LUE, JUE than direct Painlev\'e integration. The aim of this paper is not to improve on Fredholm methods, but to see to what extent one can numerically recover the \emph{correct} Painlev\'e solution from finite-$n$ data and how unstable this reconstruction is. Numerically, we verify the equality between the Fredholm- and Painlev\'e-based CDFs by combining (a) high-accuracy Nystr\"om discretizations of the finite-$n$ Fredholm determinants \cite{bornemann2010numerical} with (b) an anchored, branch-locked integration of the $\sigma$-form ODEs, where anchors are extracted from local least-squares fits to $\log\det(I-\mathsf K)$. Our results confirm agreement across GUE/LUE/JUE with precision of $O(10^{-3})$ to $O(10^{-5})$ (occasionally $O(10^{-2})$) and illustrate the finite-$n$ to scaling-limit transition. The theoretical connections to $\tau$-functions and Virasoro constraints follow the framework of \cite{adler2000random,forrester2003painleve}