In this note we first briefly review some recent progress in the study of the circularβensemble on the unit circle,whereβ>0 is a model parameter.In the special casesβ=1,2 and 4,this ensemble describes the joint ...In this note we first briefly review some recent progress in the study of the circularβensemble on the unit circle,whereβ>0 is a model parameter.In the special casesβ=1,2 and 4,this ensemble describes the joint probability density of eigenvalues of random orthogonal,unitary and sympletic matrices,respectively.For generalβ,Killip and Nenciu discovered a five-diagonal sparse matrix model,the CMV representation.This representation is new even in the caseβ=2;and it has become a powerful tool for studying the circularβensemble.We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.展开更多
基金supported by National Natural Science Foundation of China(Grant No.10671176)
文摘In this note we first briefly review some recent progress in the study of the circularβensemble on the unit circle,whereβ>0 is a model parameter.In the special casesβ=1,2 and 4,this ensemble describes the joint probability density of eigenvalues of random orthogonal,unitary and sympletic matrices,respectively.For generalβ,Killip and Nenciu discovered a five-diagonal sparse matrix model,the CMV representation.This representation is new even in the caseβ=2;and it has become a powerful tool for studying the circularβensemble.We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.
