科研之友
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摘要
We study the probability distribution function P-n((beta)) (w) of the Schmidt-like random variable w = x(1)(2)/(Sigma(n)(j=1) x(j)(2)/n), where x(j), (j = 1, 2, ... , n), are unordered eigenvalues of a given n x n beta-Gaussian random matrix, beta being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual (randomly chosen) eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such beta-Gaussian random matrices. We showthat in the asymptotic limit n -> infinity and for arbitrary beta the distribution P-n((beta)) (w) converges to the Mar. cenko-Pastur form, i. e. is defined as P-n((beta)) (w) similar to root(4 -w)/w for w is an element of [0, 4] and equals zero outside of the support, despite the fact that formally w is defined on the interval [0, n]. Furthermore, for Gaussian unitary ensembles (beta = 2) we present exact explicit expressions for P-n((beta= 2)) (w) which are valid for arbitrary n and analyse their behaviour.
- 出版日期2013-3-22
