A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions

Estelle Basor; Pavel Bleher; Robert Buckingham; Tamara Grava; Alexander Its; Elizabeth Its; Jonathan P. Keating

doi:10.1088/1361-6544/ab28c7

A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions

Estelle Basor, Pavel Bleher, Robert Buckingham, Tamara Grava, Alexander Its, Elizabeth Its, Jonathan P. Keating

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the σ-Painlevé V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the σ-Painlevé V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the σ-Painlevé III' equation. Using the conformal block expansion of the τ-functions associated with the σ-Painlevé V and the σ-Painlevé III equations leads to general conjectures for the joint moments.

Original language	English
Pages (from-to)	4033-4078
Number of pages	47
Journal	Nonlinearity
Volume	32
Issue number	10
DOIs	https://doi.org/10.1088/1361-6544/ab28c7
Publication status	Published - 13 Sep 2019

Keywords

CUE ensembles
Riemann zeta function
Riemann-Hilbert problems
Painleve equations

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10.1088/1361-6544/ab28c7

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Cite this

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title = "A representation of joint moments of CUE characteristic polynomials in terms of Painlev{\'e} functions",

abstract = "We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the σ-Painlev{\'e} V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the σ-Painlev{\'e} V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the σ-Painlev{\'e} III' equation. Using the conformal block expansion of the τ-functions associated with the σ-Painlev{\'e} V and the σ-Painlev{\'e} III equations leads to general conjectures for the joint moments.",

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T1 - A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions

AU - Basor, Estelle

AU - Bleher, Pavel

AU - Buckingham, Robert

AU - Grava, Tamara

AU - Its, Alexander

AU - Its, Elizabeth

AU - Keating, Jonathan P.

PY - 2019/9/13

Y1 - 2019/9/13

N2 - We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the σ-Painlevé V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the σ-Painlevé V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the σ-Painlevé III' equation. Using the conformal block expansion of the τ-functions associated with the σ-Painlevé V and the σ-Painlevé III equations leads to general conjectures for the joint moments.

AB - We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the σ-Painlevé V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the σ-Painlevé V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the σ-Painlevé III' equation. Using the conformal block expansion of the τ-functions associated with the σ-Painlevé V and the σ-Painlevé III equations leads to general conjectures for the joint moments.

KW - CUE ensembles

KW - Riemann zeta function

KW - Riemann-Hilbert problems

KW - Painleve equations

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DO - 10.1088/1361-6544/ab28c7

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SP - 4033

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JO - Nonlinearity

JF - Nonlinearity

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A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions

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Keywords

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LogCorRM: Log Correlations and Random Matrices

L-functions and modular forms

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A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions

Abstract

Keywords

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Other files and links

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Projects

LogCorRM: Log Correlations and Random Matrices

L-functions and modular forms

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