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New tests of the correspondence between unitary eigenvalues and the zeros of Riemann’ s zeta function
, 1999
"... This paper presents some new statistical tests and new conjectures regarding the correspondence between the eigenvalues of random unitary matrices and the zeros of Riemann’s zeta function. Global features such as the trace and number of eigenvalues in intervals are compared. Our results show satisfy ..."
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Cited by 10 (2 self)
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This paper presents some new statistical tests and new conjectures regarding the correspondence between the eigenvalues of random unitary matrices and the zeros of Riemann’s zeta function. Global features such as the trace and number of eigenvalues in intervals are compared. Our results show satisfying matchups between the two domains. They give examples of large natural datasets that follow classical distributions to high accuracy. PACS numbers: 02.10.Yn, 02.10.De 1.
On the Moments of Traces of Matrices of Classical Groups
"... We consider random matrices, belonging to the groups U(n), O(n), SO(n), and Sp(n) and distributed according to the corresponding unit Haar measure. We prove that the moments of traces of powers of the matrices coincide with the moments of certain Gaussian random variables if the order of moments is ..."
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Cited by 8 (0 self)
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We consider random matrices, belonging to the groups U(n), O(n), SO(n), and Sp(n) and distributed according to the corresponding unit Haar measure. We prove that the moments of traces of powers of the matrices coincide with the moments of certain Gaussian random variables if the order of moments is low enough. Corresponding formulas, proved partly before by various methods, are obtained here in the framework of a unique method, reminiscent of the method of correlation equations of statistical mechanics. The equations are derived by using a version of the integration by parts. 1
Deviations from the Circular Law
 Probab. Theory Related Fields 130 337– 367. MR2095933
, 2003
"... Consider Ginibre's ensemble of N N nonHermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance . As N " 1 the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane. In thi ..."
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Cited by 6 (3 self)
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Consider Ginibre's ensemble of N N nonHermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance . As N " 1 the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane. In this note we describe uctuations about this Circular Law. First we obtain nite N formulas for the covariance of certain linear statistics of the eigenvalues. Asymptotics of these objects coupled with a theorem of Costin and Lebowitz then result in central limit theorems for a variety of these statistics. 1
AVERAGES OVER CLASSICAL COMPACT LIE GROUPS AND WEYL CHARACTERS
, 2008
"... We compute EG ( ∏ i tr(gλi)), where G = Sp(2n) or SO(m) (m = 2n, 2n + 1) with Haar measure. This was first obtained by Diaconis and Shahshahani [9], but our proof is more selfcontained and gives a combinatorial description for the answer. We also consider how averages of general symmetric function ..."
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We compute EG ( ∏ i tr(gλi)), where G = Sp(2n) or SO(m) (m = 2n, 2n + 1) with Haar measure. This was first obtained by Diaconis and Shahshahani [9], but our proof is more selfcontained and gives a combinatorial description for the answer. We also consider how averages of general symmetric functions EGΦn are affected when we introduce a Weyl character χG λ into the integrand. We show that the value of EGχG λ Φn/EGΦn approaches a constant for large n. More surprisingly, the ratio we obtain only changes with Φn and λ and is independent of the Cartan type of G. Even in the unitary case, Bump and Diaconis [4] have obtained the same ratio. Finally, those ratios can be combined with asymptotics for EGΦn due to Johansson [11] and provide asymptotics for EGχG λ Φn.