A NECESSARY AND SUFFICIENT CONDITION FOR EDGE UNIVERSALITY OF WIGNER MATRICES

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In this paper, we prove a necessary and sufficient condition for the Tracy Widom law of Wigner matrices. Consider N x N symmetric Wigner matrices H with H(i)j N-(1/2)x(ij) whose upper-right entries xisi (1 <= i <f <= N) are independent and identically distributed (i.i.d.) random variables with distribution v and diagonal entries xii (1 <= i <= N) are i.i.d. random variables with distribution 13. The means of v and 15 are zero, the variance of v is 1, and the variance of 13 is finite. We prove that the Tracy Widom law holds if and only if lim(s ->infinity) s(4) P(vertical bar x(12)vertical bar >= s) = 0. The same criterion holds for Hermitian Wigner matrices.
Publisher
DUKE UNIV PRESS
Issue Date
2014-01
Language
English
Article Type
Article
Citation

DUKE MATHEMATICAL JOURNAL, v.163, no.1, pp.117 - 173

ISSN
0012-7094
DOI
10.1215/00127094-2414767
URI
http://hdl.handle.net/10203/189458
Appears in Collection
MA-Journal Papers(저널논문)
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