Motzkin path decompositions of functionals in noncommutative probability

Abstract

The decomposition of free random variables into series of orthogonal replicas gives random variables associated with important notions of noncommutative independence: boolean, monotone, orthogonal, free and free with subordination. In this paper, we study this decomposition from a new perspective. First, we show that the mixed moments of orthogonal replicas define functionals indexed by the elements of the lattices of Motzkin paths. By a duality relation, we then obtain a family of path-dependent linear functionals on the free product of algebras. They play the role of a generating set of the space of product functionals in which the boolean product corresponds to constant Motzkin paths and the free product to the sums of all Motzkin paths. Similar Motzkin path decompositions can be derived for other product functionals. In simplified terms, this paper initiates the `Motzkin path approach to noncommutative probability', in which functionals convolving variables according to different models of noncommutative independence are obtained by taking suitable subsets of the set of Motzkin paths and perhaps conditioning them on the algebra labels in nonsymmetric models. Consequently, we obtain a unified framework that allows us to look at functionals inherent to different notions of independence together rather than separately.Comment: 39 pages, 4 figure