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Communications in Mathematical Physics

, Volume 268, Issue 2, pp 403–414 | Cite as

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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