Asymptotic analysis of a Family of Painlevé Functions with Applications to CUE Derivative Moments
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| Format: | Preprint |
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2025
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| author | Bothner, Thomas Wei, Fei |
| author_facet | Bothner, Thomas Wei, Fei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_18118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic analysis of a Family of Painlevé Functions with Applications to CUE Derivative Moments Bothner, Thomas Wei, Fei Mathematical Physics Number Theory Probability Exactly Solvable and Integrable Systems Primary 33E17, Secondary 11M50, 60B20, 15B52 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. |
| title | Asymptotic analysis of a Family of Painlevé Functions with Applications to CUE Derivative Moments |
| topic | Mathematical Physics Number Theory Probability Exactly Solvable and Integrable Systems Primary 33E17, Secondary 11M50, 60B20, 15B52 |
| url | https://arxiv.org/abs/2511.18118 |