Geodesic
Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants
Documenta mathematica, Tome 12 (2007), pp. 1-70.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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".
Classification : 46L54, 15A52, 60F05
Keywords: free cumulants, random matrices, planar diagrams 
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     title = {Second order freeness and fluctuations of random matrices. {III:} {Higher} order freeness and free cumulants},
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Collins, Benoît; Mingo, James A.; Śniady, Piotr; Speicher, Roland. Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants. Documenta mathematica, Tome 12 (2007), pp. 1-70. http://geodesic.mathdoc.fr/item/DOCMA_2007__12__a18/