Applications of Realizations (aka Linearizations) to Free Probability
Applications of Realizations (aka Linearizations) to Free Probability
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实现（又名线性化）在自由概率中的应用
DOI：
10.1016/j.jfa.2017.10.003
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
R. Speicher
中科院分区：
文献类型：
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作者：
J. Helton;Tobias Mai;R. Speicher
我们展示了自由概率论中新的“线性化”思想与强大的“实现”机制的结合——在过去的50年里，在系统工程和自动机理论等领域得到了发展——如何解决随机矩阵的非交换理性函数在其大小趋于无穷时的特征值分布（甚至是非自伴随情况下的布朗测度）的确定问题。在此过程中，我们将非交换有理表达式的求值从矩阵扩展到稳定有限代数，例如II1von Neumann代数，并精确控制了有理表达式的定义域。本文提供了足够的背景信息，目的是它应该是可访问的功能分析和代数。
We show how the combination of new “linearization” ideas in free probability theory with the powerful “realization” machinery – developed over the last 50 years in fields including systems engineering and automata theory – allows solving the problem of determining the eigenvalue distribution (or even the Brown measure, in the non-selfadjoint case) of noncommutative rational functions of random matrices when their size tends to infinity. Along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras, e.g. type II1von Neumann algebras, with a precise control of the domains of the rational expressions.The paper provides sufficient background information, with the intention that it should be accessible both to functional analysts and to algebraists.