We study the probability distribution function P(β) _n(w) of the Schmidt-like random variable w = x~2 _1/(∑_j = 1~nx~2 _j), where x_j, (j = 1, 2, ..., n), are unordered eigenvalues of a given n × n β-Gaussian random matrix, β 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 β-Gaussian random matrices. We show that in the asymptotic limit n → ∞ and for arbitrary β the distribution P(β) _n(w) converges to the Mar?enko-Pastur form, i.e. is defined as for w ∈ [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 (β = 2) we present exact explicit expressions for P~((β = 2)) _n(w) which are valid for arbitrary n and analyse their behaviour.
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