Linear functionals of eigenvalues of random matrices |
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Authors: | Persi Diaconis Steven N. Evans |
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Affiliation: | Department of Mathematics, Stanford University, Building 380, MC 2125, Stanford, California 94305 ; Department of Statistics #3860, University of California at Berkeley, 367 Evans Hall, Berkeley, California 94720-3860 |
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Abstract: | Let be a random unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of to converge to a Gaussian limit as . By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of . For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices. |
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Keywords: | Random matrix central limit theorem unitary orthogonal symplectic trace eigenvalue characteristic polynomial counting function Schur function character Besov space Bessel potential |
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