非厄米特随机矩阵的特征值与非正规算子的Brown测度：厄米特约化与线性化方法
Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method
摘要：
本文研究了Voiculescu自由概率论中某些非厄米算子的Brown测度.这些算子通常表现为某些非厄米特随机矩阵集合的 *-矩的极限，而布朗测度则给出了随机矩阵极限特征值分布的一个典型候选。我们的算子的一个突出的类是由 *-自由变元中的多项式给出的。其他显式例子包括R-对角元素和椭圆元素，其中布朗测度是已知的，以及一类新的三角椭圆元素。我们的方法计算布朗措施是基于严格的数学处理的厄米减少方法，考虑在物理文献中，结合从属思想和线性化技巧。
英文摘要：
We study the Brown measure of certain non-hermitian operators arising from Voiculescu's free probability theory. Usually those operators appear as the limit in *-moments of certain ensembles of non-hermitian random matrices, and the Brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices. A prominent class for our operators is given by polynomials in *-free variables. Other explicit examples include R-diagonal elements and elliptic elements, for which the Brown measure was already known, and a new class of triangular-elliptic elements. Our method for the calculation of the Brown measure is based on a rigorous mathematical treatment of the hermitian reduction method, as considered in the physical literature, combined with subordination ideas and the linearization trick.
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