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The Instability of Painlevé Equations in Recovering Largest Eigenvalue Distributions of GUE, LUE, JUE and an Attempt of Solution to It

Haonan Gu

TL;DR

<3-5 sentence high-level summary> The paper investigates the numerical stability of recovering Painlevé-based descriptions of largest-eigenvalue distributions for GUE, LUE, and JUE from finite-n data. It compares the standard Fredholm-determinant approach with isomonodromic σ-form Painlevé descriptions, finding that direct Painlevé IV/VI integration is highly unstable and must be anchored to the Fredholm side. The authors develop an anchored, branch-locked ODE framework that interleaves sparse Painlevé dynamics with dense Fredholm data, achieving accurate reconstructions (∼10^{-3}–10^{-5}) across ensembles and edge regimes, and they extend the methodology to TW and edge limits. They also discuss limitations, sensitivity to anchoring, Hamiltonian attempts, and future RH-method directions for a more intrinsic formulation of these integrable structures.</`

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 $τ$-functions governed by Painlevé equations. For finite $n$, the associated Jimbo-Miwa-Okamoto $σ$-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 Painlevé 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} Painlevé solution from finite-$n$ data and how unstable this reconstruction is. Numerically, we verify the equality between the Fredholm- and Painlevé-based CDFs by combining (a) high-accuracy Nyström discretizations of the finite-$n$ Fredholm determinants \cite{bornemann2010numerical} with (b) an anchored, branch-locked integration of the $σ$-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 $τ$-functions and Virasoro constraints follow the framework of \cite{adler2000random,forrester2003painleve}

The Instability of Painlevé Equations in Recovering Largest Eigenvalue Distributions of GUE, LUE, JUE and an Attempt of Solution to It

TL;DR

<3-5 sentence high-level summary> The paper investigates the numerical stability of recovering Painlevé-based descriptions of largest-eigenvalue distributions for GUE, LUE, and JUE from finite-n data. It compares the standard Fredholm-determinant approach with isomonodromic σ-form Painlevé descriptions, finding that direct Painlevé IV/VI integration is highly unstable and must be anchored to the Fredholm side. The authors develop an anchored, branch-locked ODE framework that interleaves sparse Painlevé dynamics with dense Fredholm data, achieving accurate reconstructions (∼10^{-3}–10^{-5}) across ensembles and edge regimes, and they extend the methodology to TW and edge limits. They also discuss limitations, sensitivity to anchoring, Hamiltonian attempts, and future RH-method directions for a more intrinsic formulation of these integrable structures.</`

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 -functions governed by Painlevé equations. For finite , the associated Jimbo-Miwa-Okamoto -forms are (GUE), (LUE), and (JUE); under soft- or hard-edge scalings these degenerate to or 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 Painlevé 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} Painlevé solution from finite- data and how unstable this reconstruction is. Numerically, we verify the equality between the Fredholm- and Painlevé-based CDFs by combining (a) high-accuracy Nyström discretizations of the finite- Fredholm determinants \cite{bornemann2010numerical} with (b) an anchored, branch-locked integration of the -form ODEs, where anchors are extracted from local least-squares fits to . Our results confirm agreement across GUE/LUE/JUE with precision of to (occasionally ) and illustrate the finite- to scaling-limit transition. The theoretical connections to -functions and Virasoro constraints follow the framework of \cite{adler2000random,forrester2003painleve}

Paper Structure

This paper contains 31 sections, 1 theorem, 79 equations, 13 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: ensembles, kernels, Fredholm determinants, Painlevé connections
  3. Finite-$n$ PIV for the GUE gap
  4. Edge limit: from PIV to PII (Tracy--Widom)
  5. Finite-N LUE: Fredholm determinants and $\sigma$--PV
  6. Finite-N JUE: Fredholm determinants and $\sigma$--PVI
  7. Numerical verification via Painlevé II
  8. Numerical methods
  9. Results
  10. A brief comment on conditioning
  11. Numerical verification of PIV for finite-n GUE
  12. Direct IVP for the $\sigma$--form and its failure
  13. Numerical methods (ODE-based sigma-form integration)
  14. ODE formulation with signed branch
  15. Anchor data and branch selection on each interval
  16. ...and 16 more sections

Key Result

Theorem 1

Let $F_n(s)=\mathbb{P}(\lambda_{\max}\le s)$ be the largest-eigenvalue CDF of the $n\times n$ GUE with weight $e^{-x^2}$, and define $\sigma_n$ by eq:sigma-def. Then:

Figures (13)

  • Figure 1: Tracy--Widom GUE CDF $F_2(x)$ from the Painlevé II solution (solid) and the Airy-kernel Fredholm determinant (dashed).
  • Figure 2: Absolute difference between the two numerical evaluations of $F_2(x)$.
  • Figure 3: Direct ODE solutions of the $\mathrm{P_{IV}}$$\sigma$–equation compared with the finite-$n$ GUE Fredholm determinant.
  • Figure 4: Hard-edge comparison for $(a,b)=(2,3)$ at $N=300$. Top: CDF of rescaled JUE extremal eigenvalues versus the Bessel hard-edge limit at the left (top left) and right (top right) edges. Bottom: corresponding absolute errors.
  • Figure 5: LUE: Fredholm vs. locked $\sigma$--$\mathrm{PV}$ (case 1).
  • ...and 8 more figures

Theorems & Definitions (2)

  • Theorem 1: Jimbo--Miwa--Okamoto $\sigma$--form for the finite-$n$ GUE
  • Remark 1