Communications in Mathematical Physics

, Volume 268, Issue 2, pp 403–414

# A Generalization of Wigner’s Law

• Inna Zakharevich
Article

## 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
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