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Transactions of the American Mathematical Society

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Using Boolean cumulants to study multiplication and anti-commutators of free random variables


Authors: Maxime Fevrier, Mitja Mastnak, Alexandru Nica and Kamil Szpojankowski
Journal: Trans. Amer. Math. Soc. 373 (2020), 7167-7205
MSC (2010): Primary 46L54; Secondary 46L53, 05A18, 60C05, 60B20
DOI: https://doi.org/10.1090/tran/8122
Published electronically: July 28, 2020
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Abstract: We study how Boolean cumulants can be used in order to address operations with freely independent random variables, particularly in connection to the $ *$-distribution of the product of two selfadjoint freely independent random variables, and in connection to the distribution of the anti-commutator of such random variables. A key instrument in our considerations is a new combinatorial object, coloured noncrossing partitions with a structural property which we call vertical no-repeat property. As a byproduct, we obtain several results concerning enumeration of some special sets of noncrossing partitions.


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Additional Information

Maxime Fevrier
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
Email: maxime.fevrier@u-psud.fr

Mitja Mastnak
Affiliation: Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, Nova Scotia B3H 3C3, Canada
Email: mmastnak@cs.smu.ca

Alexandru Nica
Affiliation: Department of Pure Mathematics, University of Waterloo, Ontario, Canada
Email: anica@uwaterloo.ca

Kamil Szpojankowski
Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
Email: k.szpojankowski@mini.pw.edu.pl

DOI: https://doi.org/10.1090/tran/8122
Received by editor(s): September 4, 2019
Received by editor(s) in revised form: January 21, 2020
Published electronically: July 28, 2020
Additional Notes: The research of the second and third authors was supported by a Discovery Grant from NSERC, Canada.
The research of the fourth author was partially suported by NCN grant 2016/23/D/ST1/01077.
Article copyright: © Copyright 2020 American Mathematical Society