Finite- N fluctuation formulas for random matrices
T. H. Baker,Peter J. Forrester +1 more
TLDR
For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic ΣjN=1 (xj − 〈x) is computed exactly and shown to satisfy a central limit theorem asN → ∞ as mentioned in this paper.Abstract:
For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic ΣjN=1 (xj − 〈x〉) is computed exactly and shown to satisfy a central limit theorem asN → ∞. For the circular random matrix ensemble the p.d.f.’s for the statistics ½ΣjN=1 (θj −π) and − ΣjN=1 log 2 |sinθj/2| are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem asN → ∞.read more
Citations
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Universality at the edge of the spectrum in wigner random matrices
TL;DR: In this paper, it was shown that the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian matrix weakly converge to the distributions established by Tracy and Widom in G.U.O.E.
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The importance of the Selberg integral
Peter J. Forrester,S. Warnaar +1 more
TL;DR: The Selberg integral as mentioned in this paper is an n-dimensional generalization of the Euler beta integral, which was introduced by Atle Selberg and used to prove an outstanding conjecture in random matrix theory.
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The importance of the Selberg integral
TL;DR: The Selberg integral as discussed by the authors is an n-dimensional generalization of the Euler beta integral, which was introduced by Atle Selberg and has been used to prove an outstanding conjecture in random matrix theory and cases of the Macdonald conjectures.
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On the Characteristic Polynomial of a Random Unitary Matrix
TL;DR: In this paper, a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix are presented.
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The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities
TL;DR: In this paper, Soshnikov and Alexander discuss CLT for the global and local linear statistics of random matrices from classical compact groups and prove certain combinatorial identities much in the spirit of works by Kac and Spohn.
References
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Orthogonal Polynomials
TL;DR: In this paper, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed and orthogonality on the unit circle is not discussed.
Book
Analysis of Toeplitz Operators
TL;DR: Toeplitz operators arise in plenty of applications. as discussed by the authors provides a systematic introduction to the advanced analysis of block ToePlitz operators and includes both classical results and recent developments.
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Random unitary matrices
Karol Zyczkowski,Marek Kus +1 more
TL;DR: This work generates numerically random unitary matrices and shows that the statistical properties of their spectra and eigenvectors confer to the predictions of the random-matrix theory, for both CUE and COE.