| On the Spectral Distribution of Gaussian Random Matrices |
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| 引用本文： | B.Delyon J.Yao. On the Spectral Distribution of Gaussian Random Matrices[J]. 应用数学学报(英文版), 2006, 22(2): 297-312. DOI: 10.1007/s10255-006-0306-7 |
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| 作者姓名： | B.Delyon J.Yao |
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| 作者单位： | IRMAR, Universit de Rennes I, Campus de Beaulieu F-35042 Rennes Cedex, France |
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| 摘 要： | We consider the empirical spectral distribution （ESD） of a random matrix from the Gaussian Unitary Ensemble. Based on the Plancherel-Rotaeh approximation formula for Hermite polynomials, we prove that the expected empirical spectral distribution converges at the rate of O（n^-1） to the Wigner distribution function uniformly on every compact intervals [u,v] within the limiting support （-1, 1）. Furthermore, the variance of the ESD for such an interval is proved to be （πn）^-2 logn asymptotically which surprisingly enough, does not depend on the details （e.g. length or location） of the interval, This property allows us to determine completely the covariance function between the values of the ESD on two intervals.
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| 关 键 词： | 收敛率 随机矩阵 经验光谱分布 Wigner分布 |
| 收稿时间： | 2004-12-01 |
| 修稿时间： | 2004-12-012005-09-02 |
On the Spectral Distribution of Gaussian Random Matrices |
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| B. Delyon,J. Yao. On the Spectral Distribution of Gaussian Random Matrices[J]. Acta Mathematicae Applicatae Sinica, 2006, 22(2): 297-312. DOI: 10.1007/s10255-006-0306-7 |
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| Authors: | B. Delyon J. Yao |
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| Affiliation: | (1) IRMAR, Universit de Rennes I, Campus de Beaulieu, F-35042 Rennes Cedex, France |
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| Abstract: | Abstract We consider the empirical spectral distribution (ESD) of a random matrix from the Gaussian Unitary Ensemble. Based on the Plancherel-Rotach approximation formula for Hermite polynomials, we prove that the expected empirical spectral distribution converges at the rate of O(n-1) to the Wigner distribution function uniformly on every compact intervals [u, v] within the limiting support (-1, 1). Furthermore, the variance of the ESD for such an interval is proved to be (πn)-2 log n asymptotically which surprisingly enough, does not depend on the details (e. g. length or location) of the interval. This property allows us to determine completely the covariance function between the values of the ESD on two intervals. |
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| Keywords: | Convergence rate random matrices empirical spectral distribution Wigner distribution |
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