155 research outputs found
The splitting process in free probability theory
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 Moebius 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
Monotone, free, and boolean cumulants: a shuffle algebra approach
The theory of cumulants is revisited in the "Rota way", that is, by following
a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants
are considered as infinitesimal characters over a particular combinatorial Hopf
algebra. The latter is neither commutative nor cocommutative, and has an
underlying unshuffle bialgebra structure which gives rise to a shuffle product
on its graded dual. The moment-cumulant relations are encoded in terms of
shuffle and half-shuffle exponentials. It is then shown how to express
concisely monotone, free, and boolean cumulants in terms of each other using
the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.Comment: final versio
The pre-Lie structure of the time-ordered exponential
The usual time-ordering operation and the corresponding time-ordered
exponential play a fundamental role in physics and applied mathematics. In this
work we study a new approach to the understanding of time-ordering relying on
recent progress made in the context of enveloping algebras of pre-Lie algebras.
Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in
the process. Among others, we recover the noncommutative analog of the
classical Bohnenblust-Spitzer formula, and get explicit formulae for operator
products of time-ordered exponentials
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