%0 Journal Article
%T Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions
%A Zhan，Longjun
%A Blower，Gordon
%A Chen，Yang
%A Zhu，Mengkun
%X In this paper, we study the probability density function, P(c,α,β,n) dc, of the center of mass of the finite n Jacobi unitary ensembles with parameters α > −1 and β > −1; that is the probability that trM ∈ (c, c + dc), where M are n × n matrices drawn from the unitary Jacobi ensembles. We compute the exponential moment generating function of the linear statistics ∑j=1n f(xj)≔∑j=1nxj, denoted by Mf(λ,α,β,n). The weight function associated with the Jacobi unitary ensembles reads x(1 − x), x ∈ [0, 1]. The moment generating function is the n × n Hankel determinant D(λ, α, β) generated by the time-evolved Jacobi weight, namely, w(x; λ, α, β) = x(1 − x) e, x ∈ [0, 1], α > −1, β > −1. We think of λ as the time variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion, P(x, λ) = x + p(n, λ) x + ⋯ + P(0, λ), orthogonal with respect to w(x, λ, α, β) over [0, 1] play an important role. Taking the time evolution problem studied in Basor et al. [J. Phys. A: Math. Theor. 43, 015204 (2010)], with some change of variables, we obtain a certain auxiliary variable r(λ), defined by integral over [0, 1] of the product of the unconventional orthogonal polynomials of degree n and n − 1 and w(x; λ, α, β)/x. It is shown that r(2ie) satisfies a Chazy II equation. There is another auxiliary variable, denoted as R(λ), defined by an integral over [0, 1] of the product of two polynomials of degree n multiplied by w(x; λ, α, β)/x. Then Y(−λ) = 1 − λ/R(λ) satisfies a particular Painlevé V: P(α/2, − β/2, 2n + α + β + 1, 1/2). The σ function defined in terms of the λp(n, −λ) plus a translation in λ is the Jimbo-Miwa-Okamoto σ-form of Painlevé V. The continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid, gives an approximation for the moment generating function Mf(λ,α,β,n) when n is sufficiently large. Furthermore, we deduce a new expression of Mf(λ,α,β,n) when n is finite, in terms the σ function of this is a particular case of Painlevé V. 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 P(c,α,β,n) has compact support [0, n]. This result is easily extended to the β ensembles, as long as the weight w is positive and continuous over [0, 1].
%8 2018-10-01
%D 2018
%J Journal of Mathematical Physics
%V 59
%@ 0022-2488
%U http://repository.um.edu.mo/handle/10692/31637
%W UM