On the global fluctuations of block Gaussian matrices

• 成果类型: Article
• 署名作者: Diaz, Mario; Mingo, James A.; Belinschi, Serban T.
• 署名单位: Queens University - Canada; Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00925-1
• 发表日期: 2020
• 页码: 599-648
• 关键词: 2nd-order freeness; free product; clt; eigenvalues; statistics

摘要

In this paper we study the global fluctuations of block Gaussian matrices within the framework of second-order free probability theory. In order to compute the second-order Cauchy transform of these matrices, we introduce a matricial second-order conditional expectation and compute the matricial second-order Cauchy transform of a certain type of non-commutative random variables. As a by-product, using the linearization technique, we obtain the second-order Cauchy transform of non-commutative rational functions evaluated on selfadjoint Gaussian matrices.