Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions
Zhan,Longjun1; Blower,Gordon2; Chen,Yang1; Zhu,Mengkun3
Source PublicationJournal of Mathematical Physics
AbstractIn 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].
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Document TypeJournal article
CollectionUniversity of Macau
Corresponding AuthorZhu,Mengkun
Affiliation1.Department of Mathematics,University of Macau,Taipa,Avenida da Universidade,Macao
2.Department of Mathematics and Statistics,Lancaster University,Lancaster,LA14YF,United Kingdom
3.School of Science,Qilu University of Technology (Shandong Academy of Sciences),Jinan,250353,China
First Author AffilicationUniversity of Macau
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GB/T 7714
Zhan,Longjun,Blower,Gordon,Chen,Yang,et al. Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions[J]. Journal of Mathematical Physics,2018,59(10).
APA Zhan,Longjun,Blower,Gordon,Chen,Yang,&Zhu,Mengkun.(2018).Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions.Journal of Mathematical Physics,59(10).
MLA Zhan,Longjun,et al."Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions".Journal of Mathematical Physics 59.10(2018).
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