# A Generalization of Wigner’s Law

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## 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.

## Keywords

Random Matrix Regular Graph Nonzero Entry Eigenvalue Distribution Real Symmetric Matrix## Preview

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