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Distribution of Schmidt-like eigenvalues for Gaussian Ensembles of the Random Matrix Theory

We analyze the form of the probability distribution function P_{n}^{(β)}(w) of the Schmidt-like random variable w = x_1^2/(\sum_{j=1}^n x^{2}_j/n), where x_j are the eigenvalues of a given n \times n β-Gaussian random matrix, βbeing the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual 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 \to \infty and for arbitrary βthe distribution P_{n}^{(β)}(w) converges to the Marčenko-Pastur form, i.e., is defined as P_{n}^{(β)}(w) \sim \sqrt{(4 - w)/w} for w \in [0,4] and equals zero outside of the support. Furthermore, for Gaussian unitary (β= 2) ensembles we present exact explicit expressions for P_{n}^{(β=2)}(w) which are valid for arbitrary n and analyze their behavior.