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Universality of Wigner random matrices via the four moment theorem

Random Matrix Theory and Its Applications I September 13, 2010 - September 17, 2010

September 16, 2010 (09:30 AM PDT - 10:10 AM PDT)
Speaker(s): Terence Tao (University of California, Los Angeles)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC

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Abstract There has been much recent progress on understanding the fine-scale structure of the spectrum of Wigner random matrices (Hermitian random matrices whose upper-triangular entries are independent), including the important localisation results of Erdos-Schlein-Yau, and the universality results that have been obtained both through the local relaxation flow techniques of Erdos-Schlein-Yau, and through the four moment theorem of Van Vu and the speaker. In this talk we will focus on the four moment theorem, which roughly speaking asserts that the local behaviour of two random matrix ensembles are asymptotically equivalent if the entries have moments matching to fourth order. Interestingly, the need for four matching moments is necessary in some senses, but not in others, as we will describe in the talk.
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