Abstract
We present a generalization of Wigner’s semicircle law: we consider a sequence of probability distributions \((p_1,p_2,\dots)\), with mean value zero and take an N × N real symmetric matrix with entries independently chosen from p N and analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N → ∞ for certain p N the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the k th moment of p N (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when p N does not depend on N we obtain Wigner’s law: if all moments of a distribution are finite, the distribution of eigenvalues is a semicircle.
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Communicated by P. Sarnak
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Zakharevich, I. A Generalization of Wigner’s Law. Commun. Math. Phys. 268, 403–414 (2006). https://doi.org/10.1007/s00220-006-0074-5
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DOI: https://doi.org/10.1007/s00220-006-0074-5