Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices
Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices
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环形非交叉排列和划分，以及随机矩阵的二阶渐近
DOI：
10.1155/s1073792804133023
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发表时间：
2003
影响因子：
1
通讯作者：
A. Nica
中科院分区：
文献类型：
--
作者：
J. Mingo;A. Nica
我们研究了{1，.，p+q}的排列的集合Sann−nc（p，q），这些排列在一个环中是不相交的，该环的外圆上标记有p个点，内环上标记有q个点。我们通过识别可能发生在环中的交叉模式来代数地定义Sann-nc（p，q，q）。我们证明了环形对应的“测地线条件”所示的Biane在一个磁盘上的非交叉置换的特点。我们研究了Sann-nc（p，q，q）和{1，...，p+q}的环形非交叉划分的集合NC ann（p，q）之间的关系，并观察到（与圆盘的情况不同）从Sann-nc（p，q）到NC ann（p，q）的自然映射有一种病态，它阻止了它是单射的。我们指出，环形不交叉排列出现在某些家庭（Wishart和GUE）的随机矩阵的联合矩的二阶渐近的描述。一些公式扩展到一个多环框架，作为应用，我们观察到的现象的渐近高斯的痕迹的话独立的Wishart矩阵。
We study the set Sann−nc(p,q) of permutations of {1, …, p+q} which are noncrossing in an annulus with p points marked on its external circle and q points marked on its internal circle. We define Sann−nc(p,q,q) algebraically by identifying the crossing patterns which can occur in an annulus. We prove the annular counterpart for a “geodesic condition” shown by Biane to characterize noncrossing permutations in a disc. We examine the relation between Sann−nc(p,q,q) and the set NC ann (p,q of annular noncrossing partitions of {1, …, p+q} and observe that (unlike in the disc case) the natural map from Sann−nc(p,q) onto NC ann (p,q) has a pathology which prevents it from being injective. We point out that annular noncrossing permutations appear in the description of the second-order asymptotics for the joint moments of certain families (Wishart and GUE) of random matrices. Some of the formulas extend to a multiannular framework; as an application of that, we observe a phenomenon of asymptotic Gaussianity for traces of words made with independent Wishart matrices.