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MR3101841 (Review) 60B20
Tao, Terence [Tao, Terence C.] (1-UCLA)
The asymptotic distribution of a single eigenvalue gap of a Wigner matrix.
(English summary)
Probab. Theory Related Fields 157 (2013), no. 1-2, 81–106.
Let Mn be a Hermitian n × n Wigner matrix and (λi (Mn ))ni=1 the collection of its nondecreasing eigenvalues. Let ρSC stand for the density of the Wigner distribution and
ui be the real where the distribution function of ρSC equals i/n. If n ≤ i ≤ (1 − )n
for some > 0 then the author proves under certain conditions on moments that the
rescaled i-th single eigenvalue gap
λi+1 (Mn ) − λi (Mn )
√
(1/ nρSC (ui ))
converges in distribution as n → ∞ to the so-called Gaudin-Mehta distribution. The
proof is written for the GUE and the result extends to Wigner matrices by the Four
Nizar Demni
Moment Theorem due to the author and Van Vu.
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