Free probability for purely discrete eigenvalues of random matrices
Free probability for purely discrete eigenvalues of random matrices
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DOI：
10.2969/jmsj/77147714
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发表时间：
2015-12
影响因子：
0.7
通讯作者：
B. Collins;Takahiro Hasebe;Noriyoshi Sakuma
中科院分区：
文献类型：
--
作者：
B. Collins;Takahiro Hasebe;Noriyoshi Sakuma
本文研究了两类随机矩阵变量的非交换多项式形式的随机矩阵模型：（a）第一类在极限下具有离散谱，（B）第二类在Voiculescu意义下具有联合极限分布且全局旋转不变。我们假设构成这个多项式的每个单项式至少包含一个类型（a）的变量，并证明这个随机矩阵模型有一组特征值，几乎肯定收敛到一组确定的数字，这组数字要么是有限的，要么在大尺寸极限下只累积到零。为此，我们定义了一个框架（循环单调独立）分析离散谱和发展的时刻方法的特征值紧凑（特别是Schatten类）运营商。我们给出了几个明确的计算我们的模型的离散特征值。
In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.