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The Splitting Process in Free Probability Theory.
- Source :
-
IMRN: International Mathematics Research Notices . 2016, Vol. 2016 Issue 9, p2647-2676. 30p. - Publication Year :
- 2016
-
Abstract
- Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Möbius calculus over the lattice of non-crossing partitions. In this work, we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed-point equation that relates it to the generating series of free cumulants. These phenomena are explained through a process similar to (though different from) the arborification process familiar in the theory of dynamical systems, and originating in Cayley's work. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 10737928
- Volume :
- 2016
- Issue :
- 9
- Database :
- Academic Search Index
- Journal :
- IMRN: International Mathematics Research Notices
- Publication Type :
- Academic Journal
- Accession number :
- 119909051
- Full Text :
- https://doi.org/10.1093/imrn/rnv209