Star-cumulants of free unitary Brownian motion

We study joint free cumulants of ut and ut⁎, where ut is a free unitary Brownian motion at time t. We determine explicitly some special families of such cumulants. On the other hand, for a general joint cumulant of ut and ut⁎, we “calculate the derivative” for t→∞, when ut approaches a Haar unitary....

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Bibliographic Details
Published in:Advances in applied mathematics 2015-08, Vol.69, p.1-45
Main Authors: Demni, Nizar, Guay-Paquet, Mathieu, Nica, Alexandru
Format: Article
Language:English
Subjects:
Combinatorics
Free cumulants
Free probability
Free unitary Brownian motion
Infinitesimal determining sequence
Mathematics
Non-crossing partitions
Operator Algebras
Probability
R-diagonal element
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Summary:We study joint free cumulants of ut and ut⁎, where ut is a free unitary Brownian motion at time t. We determine explicitly some special families of such cumulants. On the other hand, for a general joint cumulant of ut and ut⁎, we “calculate the derivative” for t→∞, when ut approaches a Haar unitary. In connection to the latter calculation we put into evidence an “infinitesimal determining sequence” which naturally accompanies an arbitrary R-diagonal element in a tracial ⁎-probability space.
ISSN:0196-8858
1090-2074
DOI:10.1016/j.aam.2015.05.001