The asymptotic distribution of a single eigenvalue gap of a Wigner matrix

• 成果类型: Article
• 署名作者: Tao, Terence
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0450-3
• 发表日期: 2013
• 页码: 81-106
• 关键词: universality; ensembles

摘要

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.