Finite free convolutions via Weingarten calculus

IF 0.9 4区数学 Q4 PHYSICS, MATHEMATICAL

Random Matrices-Theory and Applications Pub Date : 2019-07-01 DOI:10.1142/S2010326321500386

J. Campbell, Z. Yin

引用次数: 2

Abstract

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 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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通过Weingarten微积分得到的有限自由卷积

我们考虑Marcus, Spielman和Srivastava在最近的一篇论文中研究的多项式的三个有限自由卷积。每个都可以用直接的显式公式来描述，也可以用随机旋转矩阵的运算来描述。我们提出了一种替代的方法来等价于这些描述，基于组合Weingarten方法的积分在酉群和正交群。我们的方法的一个关键方面是确定一定的正交性质，该性质由酉群的一些重要的子群(包括酉、正交和有符号置换矩阵的群)满足，并产生所需的卷积公式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Random Matrices-Theory and Applications Decision Sciences-Statistics, Probability and Uncertainty

CiteScore

1.90

自引率

11.10%

发文量

期刊介绍： Random Matrix Theory (RMT) has a long and rich history and has, especially in recent years, shown to have important applications in many diverse areas of mathematics, science, and engineering. The scope of RMT and its applications include the areas of classical analysis, probability theory, statistical analysis of big data, as well as connections to graph theory, number theory, representation theory, and many areas of mathematical physics. Applications of Random Matrix Theory continue to present themselves and new applications are welcome in this journal. Some examples are orthogonal polynomial theory, free probability, integrable systems, growth models, wireless communications, signal processing, numerical computing, complex networks, economics, statistical mechanics, and quantum theory. Special issues devoted to single topic of current interest will also be considered and published in this journal.