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

Zhan, Longjun and Blower, Gordon and Chen, Yang and Zhu, Mengkun (2018) Center of mass distribution of the Jacobi unitary ensembles:Painleve V, asymptotic expansions. Journal of Mathematical Physics, 59 (10). ISSN 0022-2488

PDF (PVpaper)
PVpaper.pdf - Accepted Version
Available under License Creative Commons Attribution-NonCommercial.

Download (19MB)


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 ${\rm tr}M_n\in(c, c+dc),$ 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 $\mathcal{M}_f(\lambda,\alpha,\beta,n)$. The weight function associated with the Jacobi unitary ensembles reads $x^{\alpha}(1-x)^{\beta},\; x\in [0,1]$. The moment generating function is the $n\times n$ Hankel determinant $D_n(\lambda,\alpha,\beta)$ generated by the time-evolved Jacobi weight, namely, $w(x;\lambda ,\alpha,\beta )=x^{\alpha}(1-x)^{\beta}\,{\rm e}^{-\lambda\:x},\,x\in[0,1],\,\alpha>-1,\,\beta>-1$. We think of $\lambda$ as the time variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion, $P_n(x,\lambda)= x^n+ p(n,\lambda)\:x^{n-1}+\dots+P_n(0,\lambda)$, orthogonal with respect to $w(x,\lambda,\alpha,\beta )$ over $[0,1]$ play an important role. Taking the time evolution problem studied in Basor, Chen and Ehrhardt (\cite{BasorChenEhrhardt2010}), with some change of variables, we obtain a certain auxiliary variable $r_n(\lambda),$ defined by integral over $[0,1]$ of the product of the unconventional orthogonal polynomials of degree $n$ and $n-1$ and $w(x,\lambda,\alpha,\beta )/x$. It is shown that $r_n(2\imath\/{\rm e}^{z})$ satisfies a Chazy $II$ equation. There is another auxiliary variable, denote as $R_n(\lambda),$ defined by an integral over $[0,1]$ of the product of two polynomials of degree $n$ multiplied by $w(x,\lambda)/x.$ Then $Y_n(-\lambda)=1-\lambda/R_n(\lambda)$ satisfies a particular Painlev\'{e} \uppercase\expandafter{\romannumeral 5}: $P_{\rm V}(\alpha^2/2$, $ -\beta^2/2, 2n+\alpha+\beta+1,1/2)$.\\ The $\sigma_n$ function defined in terms of the $\lambda\:p(n,-\lambda)$ plus a translation in $\lambda$ is the Jimbo--Miwa--Okamoto $\sigma$-form of Painlev\'{e} \uppercase\expandafter{\romannumeral 5}. In the continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid, gives an approximation for the moment generation function $\mathcal{M}_f(\lambda,\alpha,\beta,n)$ when $n$ is sufficiently large. Furthermore, we deduce a new expression of $\mathcal{M}_f(\lambda,\alpha,\beta,n)$ when $n$ is finite in terms the $\sigma$ function of this the Painlev\'{e} \uppercase\expandafter{\romannumeral 5} An estimate shows that the moment generating function is a function of exponential type and of order $n$. From the Paley-Wiener theorem, one deduces that $\mathbb{P}(c,\alpha,\beta,n)$ has compact support $[0,n]$. This result is easily extended to the $\beta$ ensembles, as long as $w$ the weight is positive and continuous over $[0,1].$

Item Type:
Journal Article
Journal or Publication Title:
Journal of Mathematical Physics
Additional Information:
Copyright 2018 American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 59 (10), 2018 and may be found at This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.
Uncontrolled Keywords:
ID Code:
Deposited By:
Deposited On:
21 Aug 2018 14:26
Last Modified:
30 Sep 2020 08:05