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Asymptotic analysis of a Family of Painlevé Functions with Applications to CUE Derivative Moments

Thomas Bothner, Fei Wei

TL;DR

This work develops a unifying Riemann–Hilbert framework connecting the joint moments of CUE characteristic polynomial derivatives to Painlevé transcendents. By representing $F_N(s,h)$ as a Hankel determinant linked to confluent hypergeometric functions and performing a detailed large-$N$ RH analysis, the authors obtain a σ-Painlevé III$'$ description for the limiting function $F(s,h)$ across real $s>-1/2$ and complex $h$. They derive integral representations of $F(s,h)$ in terms of a Painlevé-III$'$ solution $v(z;s)$ and prove a density for the Hua–Pickrell random variable $X(s)$, answering a question of ABGS. The results connect leading coefficients of joint moments to integrable systems and provide a rigorous pathway to studying mean values of the Hardy $Z$-function, offering a new bridge between random matrix theory, Painlevé equations, and analytic number theory.

Abstract

The Riemann-Hilbert method is employed to carry out an asymptotic analysis of a family of $σ$-Painlevé V functions associated with Hankel determinants involving the confluent hypergeometric function of the second kind. In the large-matrix limit, this family degenerates to a family of $σ$-Painlevé III$'$ functions, whose precise asymptotic behavior is also obtained. Both families of Painlevé functions arise in the study of the joint moments of the derivative of the characteristic polynomial of a CUE random matrix and the polynomial itself, whose asymptotics are closely related to the moments of the Riemann zeta function and the Hardy $\mathsf{Z}$-function on the critical line. One of our main results establishes a representation of the leading coefficients of these joint moments in terms of $σ$-Painlevé III$'$ functions for general real exponents. The other main result resolves a question of Assiotis et al. in [Probab. Math. Physics. 2(2021), 613-642, Remark 2.5] concerning the existence of a probability density for a random variable arising in the ergodic decomposition of Hua-Pickrell measures.

Asymptotic analysis of a Family of Painlevé Functions with Applications to CUE Derivative Moments

TL;DR

This work develops a unifying Riemann–Hilbert framework connecting the joint moments of CUE characteristic polynomial derivatives to Painlevé transcendents. By representing as a Hankel determinant linked to confluent hypergeometric functions and performing a detailed large- RH analysis, the authors obtain a σ-Painlevé III description for the limiting function across real and complex . They derive integral representations of in terms of a Painlevé-III solution and prove a density for the Hua–Pickrell random variable , answering a question of ABGS. The results connect leading coefficients of joint moments to integrable systems and provide a rigorous pathway to studying mean values of the Hardy -function, offering a new bridge between random matrix theory, Painlevé equations, and analytic number theory.

Abstract

The Riemann-Hilbert method is employed to carry out an asymptotic analysis of a family of -Painlevé V functions associated with Hankel determinants involving the confluent hypergeometric function of the second kind. In the large-matrix limit, this family degenerates to a family of -Painlevé III functions, whose precise asymptotic behavior is also obtained. Both families of Painlevé functions arise in the study of the joint moments of the derivative of the characteristic polynomial of a CUE random matrix and the polynomial itself, whose asymptotics are closely related to the moments of the Riemann zeta function and the Hardy -function on the critical line. One of our main results establishes a representation of the leading coefficients of these joint moments in terms of -Painlevé III functions for general real exponents. The other main result resolves a question of Assiotis et al. in [Probab. Math. Physics. 2(2021), 613-642, Remark 2.5] concerning the existence of a probability density for a random variable arising in the ergodic decomposition of Hua-Pickrell measures.

Paper Structure

This paper contains 33 sections, 49 theorems, 340 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A brief history of $F(s,h)$
  3. $F(s, h)$ in terms of a Painlevé function for general exponents $s$ and $h$.
  4. Comparison with previous results
  5. Riemann-Hilbert approach
  6. A representation of $F_{N}(s,h)$ in terms of a Hankel determinant
  7. Large $N$-asymptotic of $u_{N}(z;s)$
  8. Large $z$-asymptotic of $v(z;s)$
  9. Small $z$-asymptotic of $v(z;s)$
  10. An integral representation of $F(s,h)$ for general exponents $s$ and $h$
  11. A question of Assiotis, Bedert, Gunnes and Soor posed in ABGS
  12. Preliminary results
  13. Large $N$-asymptotics for $J_N(\xi,s)$
  14. The outer parametrix
  15. The parametrix at $\lambda=0$
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $s > -\tfrac{1}{2}$ and $h \in \mathbb{C}$ with $0 \leq \mathrm{Re}(h) < \tfrac{1}{2} + s$. Then object holds with where $G(z)$ is the Barnes' $G$-function, cf. NIST, $\Gamma(z)$ Euler's Gamma function, cf. NIST, and $M\in\mathbb{Z}_{\geq 0}$ such that $\mathrm{Re}(h) \in (M, M+1)$. The function $v(z)=v(z; s)$ satisfies the $\sigma$-Painlevé III$'$ equation, cf. O, and it has the explicit fo

Figures (1)

  • Figure 1: The oriented jump contour $(-\infty,-z)\cup(-z,0)\cup(0,\infty)$ used in the RHP for $\Phi(\lambda)$, in the complex $\lambda$-plane drawn in red. The (end)points $\lambda=0,-z$ are colored in blue, for one particular choice $z>0$, and the branch cut is oriented to the right.

Theorems & Definitions (98)

  • Theorem 1.1
  • Conjecture 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Corollary 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Lemma 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 88 more