A representation of joint moments of CUE characteristic polynomials in terms of Painleve 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 sigma-Painleve 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 sigma-Painleve 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 sigma-Painleve III equation. Using the conformal block expansion of the tau-functions associated with the sigma-Painleve V and the sigma-Painleve III equations leads to general conjectures for the joint moments.

39 pages. To appear in Nonlinearity

References in corpus (3)

• Irregular conformal blocks and connection formulae for Painlevé V functions
• Mixed moments of characteristic polynomials of random unitary matrices
• A Combinatorial Approach to Mixed Ratios of Characteristic Polynomials

Cited by in corpus (9)

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• Emptiness formation probability and Painlevé V equation in the XY spin chain
• Exact equivalences and phase discrepancies between random matrix ensembles
• Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble
• Rational Solutions of the Fifth Painlevé Equation. Generalised Laguerre Polynomials
• Convergence and an explicit formula for the joint moments of the Circular Jacobi $\beta$-Ensemble characteristic polynomial
• Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices
• On a distinguished family of random variables and Painlevé equations
• Schur expansion of random-matrix reproducing kernels