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.
SU ZhongGen Department of Mathematics,Zhejiang University,Hangzhou 310027,China
让 { X， X 1， X 2， ...} 是零个平均数严格地静止 ? 混合顺序。集合 S n = 危 k=1 n X k 和 f (x p )= 危 n=1 鈭 ? n r ? 2P (| S n |鈮 ? x p ) 。什么时候？>() 1/p，为 p > 1/2 和 r > 1，为鈭的条件？？鈭 ? f (x p dx < 保持的鈭 ? 被建立由和强壮的近似使用联合方法，它与传统的使相称和 Hoffman-J 不同 ? rgensen 不平等。关键词完全的集中 - 混合序列 - 强壮的近似先生(2000 ) 题目分类 60F15 由中国的国家自然科学基础支持了(资助 Nos. 10771192 并且 10671176 )
让是最小的 Erd ? 1 的 s-Szekeres 排列， 2， ... ， n 2，并且让 l n， k 是在片断的最长的增加随后的长度((1 ) ， ... ，(k)) 。在制服下面，我们建立指数地腐烂的措施为 l n 上面的尾巴概率跳了， k，并且作为后果，我们获得完全的集中，它是 Romiks 的改进最近的结果。我们也为 l n 给一条精确更低的指数的尾巴， k。
In this paper, we study the compound binomial model in Markovian environment, which is proposed by Cossette, et al. (2003). We obtain the recursive formula of the joint distributions of T,X(T-1) and |X(T)|(i.e., the time of ruin, the surplus before ruin and the deficit at ruin) by the method of mass function of up-crossing zero points, as given by Liu and Zhao (2007). By using the same method, the recursive formula of supremum distribution is obtained. An example is included to illustrate the results of the model.
