Duke Mathematical Journal

A necessary and sufficient condition for edge universality of Wigner matrices

Ji Oon Lee and Jun Yin

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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.

Article information

Source
Duke Math. J., Volume 163, Number 1 (2014), 117-173.

Dates
First available in Project Euclid: 8 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1389190326

Digital Object Identifier
doi:10.1215/00127094-2414767

Mathematical Reviews number (MathSciNet)
MR3161313

Zentralblatt MATH identifier
1296.60007

Subjects
Primary: 81V70: Many-body theory; quantum Hall effect
Secondary: 82C10: Quantum dynamics and nonequilibrium statistical mechanics (general)

Citation

Lee, Ji Oon; Yin, Jun. A necessary and sufficient condition for edge universality of Wigner matrices. Duke Math. J. 163 (2014), no. 1, 117--173. doi:10.1215/00127094-2414767. https://projecteuclid.org/euclid.dmj/1389190326


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