Convergence and an explicit formula for the joint moments of the Circular Jacobi $\beta$-Ensemble characteristic polynomial

Abstract

The problem of convergence of the joint moments, which depend on two parameters $s$ and $h$, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes. Recently, Forrester considered the analogous problem for the Circular $\beta$-Ensemble (C$\beta$E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer $s$ and complex $h$. In this paper we consider this problem for a generalisation of the C$\beta$E, the Circular Jacobi $\beta$-ensemble (CJ$\beta$E), depending on an additional complex parameter $\delta$ and we prove convergence of the joint moments for general positive real exponents $s$ and $h$. We give a representation for the limit in terms of the moments of a family of real random variables of independent interest. This is done by making use of some general results on consistent probability measures on interlacing arrays. Using these techniques, we also extend Forrester's explicit formula to the case of real $s$ and $\delta$ and integer $h$. Finally, we prove an analogous result for the moments of the logarithmic derivative of the characteristic polynomial of the Laguerre $\beta$-ensemble.

arXiv commentRevised according to referee reports. To appear in Mathematical Physics, Analysis and Geometry

References in corpus (16)

• Matrix Models for Beta Ensembles
• Infinite random matrices and ergodic measures
• Painlevé V and time dependent Jacobi polynomials
• Hua type integrals over unitary groups and over projective limits of unitary groups
• Gap probabilities in the finite and scaled Cauchy random matrix ensembles
• Diffusion at the random matrix hard edge
• Matrix models for circular ensembles
• Boundary conditions associated with the Painlevé III' and V evaluations of some random matrix averages
• Derivative moments for characteristic polynomials from the CUE
• The Sine${}_{\beta }$ operator
• A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions
• Mixed moments of characteristic polynomials of random unitary matrices
• Ergodic Decomposition for Inverse Wishart Measures on Infinite Positive-Definite Matrices
• Infinite Random Matrices and Ergodic decomposition of Finite or Infinite Hua-Pickrell measures
• On a distinguished family of random variables and Painlevé equations
• Operator level limit of the circular Jacobi $\beta$-ensemble

Cited by in corpus (2)

• Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices
• Hua-Pickrell diffusions and differential equations related with pseudo-Jacobi polynomials