Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants

By Benoît Collins, James A. Mingo, Piotr Śniady, Roland Speicher. Documenta Mathematica, 2006

136 citations recorded by the source corpus.

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".

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

View Paper: https://www.semanticscholar.org/paper/e9ae446f19a5c6cb81a9ca538d55243491fe873f

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

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Citations in the corpus

Free Probability for Pairs of Faces I (2013, background): “Bi-freeness may be related to a number of recent non-commutative independencies ([2], [1], [5], [6], [7], [11]).”
Free Probability and Random Matrices (2014, background): “Extensions of the theory to rectangular matrices can be found in [11], and to “second order freeness” (describing ﬂuctuations of random matrices) in [16].”
A Linearization of Connes' Embedding Problem (2007, background): “These are asymptotically Gaussian and provide examples of the phenomenon of second-order freeness, which has been treated in a recent series of papers [16], [15], [8].”
Asymptotics of unitary and orthogonal matrix integrals (2006, background): “It is not obvious to us how the Schwinger-Dyson equation can be read oﬀ from the results of [10] (without writing a change of variable invariance formula), and it would be interesting to attempt to ﬁgure out the meaning of Schwinger-Dyson equation at the representation theoretic level.”
Statistical eigen-inference from large Wishart matrices (2007, background): “For the definition of free cumulants and more information on secondorder freeness, the interested reader should consult Collins et al. (2007), in particular, Section 2.”