Journal
PROBABILITY THEORY AND RELATED FIELDS
Volume 157, Issue 1-2, Pages 81-106Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00440-012-0450-3
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Funding
- NSF [DMS-0649473]
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We show that the distribution of (a suitable rescaling of) a single eigenvalue gap of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin-Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin-Mehta law required either an averaging in the eigenvalue index parameter , or fixing the energy level instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function (where is a suitably rescaled version of ) with the event that there is no spectrum in an interval , in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.
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