基于Weingarten演算的有限自由卷积
Finite free convolutions via Weingarten calculus
摘要：
我们考虑三个有限的自由卷积多项式研究在最近的论文马库斯，斯皮尔曼，和Srivastava。每一个都可以通过直接的显式公式或随机旋转矩阵的运算来描述。我们提出了一种替代方法，这些描述之间的等价性，基于组合Weingarten方法的一体化的酉和正交群。我们的方法的一个关键方面是要确定一个特定的\emph{quadrature property}，这是由一些重要的一系列的酉群（包括酉，正交和签署的置换矩阵的群体）的子群满足，并产生所需的卷积公式。
英文摘要：
We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman, and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal groups. A key aspect of our approach is to identify a certain \emph{quadrature property}, which is satisfied by some important series of subgroups of the unitary groups (including the groups of unitary, orthogonal, and signed permutation matrices), and which yields the desired convolution formulae.
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