15 January 2014 A necessary and sufficient condition for edge universality of Wigner matrices
Ji Oon Lee, Jun Yin
Duke Math. J. 163(1): 117-173 (15 January 2014). DOI: 10.1215/00127094-2414767

Abstract

In this paper, we prove a necessary and sufficient condition for the Tracy–Widom law of Wigner matrices. Consider N×N symmetric Wigner matrices H with Hij=N1/2xij whose upper-right entries xij (1i<jN) are independent and identically distributed (i.i.d.) random variables with distribution ν and diagonal entries xii (1iN) are i.i.d. random variables with distribution ν̃. The means of ν and ν̃ are zero, the variance of ν is 1, and the variance of ν̃ is finite. We prove that the Tracy–Widom law holds if and only if lim ss4P(|x12|s)=0. The same criterion holds for Hermitian Wigner matrices.

Citation

Download Citation

"A necessary and sufficient condition for edge universality of Wigner matrices." Duke Math. J. 163 (1) 117 - 173, 15 January 2014. https://doi.org/10.1215/00127094-2414767

Information

Published: 15 January 2014
First available in Project Euclid: 8 January 2014

Digital Object Identifier: 10.1215/00127094-2414767

Vol.163 • No. 1 • 15 January 2014
Back to Top