DOCUMENTA MATHEMATICA, Vol. 12 (2007), 1-70

Benoît Collins, James A. Mingo, Piotr ´Sniady, Roland Speicher

Second Order Freeness and Fluctuations of Random Matrices III. Higher order freeness and free cumulants

We extend the relation between random matrices and free probability theory from the level of expectations to the level of all correlation functions (which are classical cumulants of traces of products of the matrices). We introduce the notion of «higher order freeness" and develop a theory of corresponding free cumulants. We show that two independent random matrix ensembles are free of arbitrary order if one of them is unitarily invariant. We prove R-transform formulas for second order freeness. Much of the presented theory relies on a detailed study of the properties of «partitioned permutations".

2000 Mathematics Subject Classification: 46L54 (Primary), 15A52, 60F05

Keywords and Phrases: free cumulants, random matrices, planar diagrams

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