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 pN and analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N → ∞ for certain pN 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 kth moment of pN (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when pN 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.
Random Matrix Regular Graph Nonzero Entry Eigenvalue Distribution Real Symmetric Matrix
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