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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
