Journal ArticleDOI
Exact results and universal asymptotics in the Laguerre random matrix ensemble
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For various values of the parameters in the Laguerre random matrix ensemble, the distribution of the smallest eigenvalue and the scaled n-level distribution function are calculated exactly in terms of generalized hypergeometric functions as mentioned in this paper.Abstract:
For various values of the parameters in the Laguerre random matrix ensemble, the distribution of the smallest eigenvalue and the scaled n‐level distribution function are calculated exactly in terms of generalized hypergeometric functions. In certain cases these functions are expressed as multidimensional integrals, from which asymptotic formulas are calculated and predictions of universal behavior verified.read more
Citations
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The Calogero-Sutherland Model and Generalized Classical Polynomials
T. H. Baker,Peter J. Forrester +1 more
TL;DR: In this article, generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of Schrodinger operators for Calogero-Sutherland type quantum systems.
Journal ArticleDOI
Application of the τ-Function Theory¶of Painlevé Equations to Random Matrices:¶PIV, PII and the GUE
TL;DR: In this article, the Okamoto τ-function theory of PIV and PII has been applied to the evaluation of the largest eigenvalue of the finite and scaled infinite GUE.
Journal ArticleDOI
The Calogero-Sutherland Model and Generalized Classical Polynomials
T. H. Baker,Peter J. Forrester +1 more
TL;DR: In this article, generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schrodinger operators for Calogero-Sutherland-type quantum systems.
Journal ArticleDOI
Random Matrices: the Distribution of the Smallest Singular Values
Terence Tao,Van Vu +1 more
TL;DR: In this article, it was shown that the probability distribution of the bottom k singular values of a real-valued random variable of mean zero and variance 1 converges to the same limiting distribution as in the special case when ξ is real gaussian.
Journal ArticleDOI
Asymptotics of Tracy-Widom Distributions and the Total Integral of a Painlevé II Function
TL;DR: In this article, the authors express the GUE Tracy-Widom distribution functions in terms of integrals starting from minus infinity, and show that these integrals can be expressed as the total integral of the Hastings-McLeod solution of the Painleve II equation.
References
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Journal ArticleDOI
Eigenvalues and condition numbers of random matrices
TL;DR: For real or complex matrices with elements from a standard normal distribution, the condition number should be given given a random matrix, and as mentioned in this paper showed that condition number is not the right condition number for any real matrix.
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Some combinatorial properties of Jack symmetric functions
TL;DR: In this article, the dominance ordering is defined as the reverse lexicographic order of the partial ordering < the natural ordering, and it is defined in terms of an infinite set of indeterminates.
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The spectrum edge of random matrix ensembles
TL;DR: In this article, the scaled n-level distribution and scaled level spacing distribution for the small and large eigenvalues of various ensembles of random matrices are considered, and exact results for both these quantities are obtained for various special values of the parameters in the gaussian and Laguerre ensemble.
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Bessel functions of matrix argument
TL;DR: In this paper, a large number of formulae from the classical theory of special functions are given appropriate generalizations, some of which turn out to have applications to lattice-point problems and to the theory of non-central Wishart distributions in statistics.
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Selberg integrals and hypergeometric functions associated with Jack polynomials
TL;DR: In this paper, a new class of hypergeometric functions of several variables is introduced by using Jack polynomials and a multivariate generalization of Aomoto's result is given.