Circular β ensembles,CMV representation,characteristic polynomials  

作　　者：SU ZhongGen Department of Mathematics,Zhejiang University,Hangzhou 310027,China 

出　　处：《Science China Mathematics》2009年第7期1467-1477,共11页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No.10671176)

摘　　要：In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle,where β > 0 is a model parameter.In the special cases β = 1,2 and 4,this ensemble describes the joint probability density of eigenvalues of random orthogonal,unitary and sympletic matrices,respectively.For general β,Killip and Nenciu discovered a five-diagonal sparse matrix model,the CMV representation.This representation is new even in the case β = 2;and it has become a powerful tool for studying the circular β ensemble.We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle, where β > 0 is a model parameter. In the special cases β = 1,2 and 4, this ensemble describes the joint probability density of eigenvalues of random orthogonal, unitary and sympletic matrices, respectively. For general β, Killip and Nenciu discovered a five-diagonal sparse matrix model, the CMV representation. This representation is new even in the case β = 2; and it has become a powerful tool for studying the circular β ensemble. We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.

关 键 词：characteristic polynomials circular β ensembles CMV representation moment identity 15A52 

分 类 号：O151.21[理学—数学]