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

By James A. Mingo, Piotr Śniady, Roland Speicher. 2004

154 citations recorded by the source corpus.

Abstract 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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/91001dd469958e7a732260871b591a94185a0af7

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Free Probability and Random Matrices (2014, methodology): “The following theorem from [128] follows directly from Definition 24 of second order freeness.”
Asymptotics of unitary and orthogonal matrix integrals (2006, background): “It is interesting to mention here that papers [24] and [25] have developed a calculus on annuli which seems to be related to our graphical model.”