Second order freeness and fluctuations of random matrices: II. Unitary random matrices

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• 作者：James A. Mingo ; Piotr Ś ; niady ; Rol ; Speicher
• 关键词：Free probability ; Haar distributed unitary random matrices ; Second order freeness
• 刊名：Advances in Mathematics
• 出版年：2007
• 出版时间：15 February 2007
• 年：2007
• 卷：209
• 期：1
• 页码：212-240
• 全文大小：260 K

文摘

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We show how the concept of “second order freeness”, which was introduced in Part I, allows one to understand global fluctuations of Haar distributed unitary random matrices. In particular, independence between the unitary ensemble and another ensemble goes in the large N limit over into asymptotic second order freeness. Two important consequences of our general theory are: (i) we obtain a natural generalization of a theorem of Diaconis and Shahshahani to the case of several independent unitary matrices; (ii) we can show that global fluctuations in unitarily invariant multi-matrix models are not universal.