Center of mass distribution of the Jacobi unitary ensembles: Painleve V, asymptotic expansions

Abstract

In this paper, we study the probability density function, $\mathbb{P}(c,\alpha ,\beta ,n)dc$, of the center of mass of the finite $n$ Jacobi unitary ensembles with parameters $\alpha >-1$ and $\beta >-1$; that is the probability that $\mathrm{tr}{M}_n\in (c,c+dc),$ where $M_n$ are $n\times n$ matrices drawn from the unitary Jacobi ensembles. We first compute the exponential moment generating function of the linear statistics ${\sum }_{j=1}^{n}f(x_j):={\sum }_{j=1}^n{x}_j,$ denoted by ${\mathcal{M}}_f(\lambda ,\alpha ,\beta ,n)$.

References in corpus (8)

• Fredholm Determinants, Differential Equations and Matrix Models
• Painlevé V and time dependent Jacobi polynomials
• Painleve IV and degenerate Gaussian Unitary Ensembles
• Nonlinear statistics of quantum transport in chaotic cavities
• Sums of divisor functions in $F_q[t]$ and matrix integrals
• Perturbed Hankel Determinants
• Linear Statistics of Matrix Ensembles in Classical Background
• On the variance of linear statistics of Hermitian random matrices

Cited by in corpus (3)

• Orthogonal Polynomials, Asymptotics and Heun Equations
• Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel
• Painlevé IV, $\sigma -$Form and the Deformed Hermite Unitary Ensembles