Abstract
Let β > 0 and consider an n -point process λ1, λ2,..., λn from Hermite β ensemble on the real line R. Dumitriu and Edelman discovered a tri-diagonal matrix model and established the global Wigner semicircle law for normalized empirical measures. In this paper, we prove that the average number of states in a small interval in the bulk converges in probability when the length of the interval is larger than √log n , i.e., local semicircle law holds. And the number of positive states in (0,∞) is proved to uctuate normally around its mean n /2 with variance like log n / π 2 β . The proofs rely largely on the way invented by Valko and Virag of counting states in any interval and the classical martingale argument.
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.
