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Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions. (English) Zbl 1402.60012

Summary: 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 \(\operatorname{tr}M_{n} \in (c, c + dc)\), where \(M_{n}\) are \(n \times n\) matrices drawn from the unitary Jacobi ensembles. We 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, \(w(x; \lambda, \alpha, \beta) = x^{\alpha}(1 - x)^{\beta} e^{-\lambda x},\; x \in [0, 1],\; \alpha > -1,; \beta > -1\). We think of \(\lambda\) as the time variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion, \(P_{n}(x, \lambda) = x^{n} + p(n, \lambda) x^{n - 1} + \cdots + P_{n}(0, \lambda)\), orthogonal with respect to \(w(x, \lambda, \alpha, \beta)\) over \([0, 1]\) play an important role. Taking the time evolution problem studied in E. Basor et al. [J. Phys. A, Math. Theor. 43, No. 1, Article ID 015204, 25 p. (2010; Zbl 1202.33030)], 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 \(r_{n}(2ie^{z})\) satisfies a Chazy II equation. There is another auxiliary variable, denoted as \(R_{n}(\lambda)\), defined by an integral over \([0, 1]\) of the product of two polynomials of degree \(n\) multiplied by \(w(x; \lambda, \alpha, \beta)/x\). Then \(Y_{n}(-\lambda) = 1 - \lambda/R_{n}(\lambda)\) satisfies a particular Painlevé V: \(P_{V}(\alpha^{2}/2, - \beta^{2}/2, 2n + \alpha + \beta + 1, 1/2)\). 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 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 \(\mathcal{M}_f(\lambda, \alpha, \beta, n)\) 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 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 \(\mathbb{P}(c, \alpha, \beta, n)\) has compact support \([0, n]\). This result is easily extended to the \(\beta\) ensembles, as long as the weight \(w\) is positive and continuous over \([0, 1]\).{
©2018 American Institute of Physics}

MSC:

60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
76W05 Magnetohydrodynamics and electrohydrodynamics
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
54E05 Proximity structures and generalizations
33E17 Painlevé-type functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Citations:

Zbl 1202.33030
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References:

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