Combinatorics of NC-Probability Spaces with Independent Constants

Abstract

Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants.In the scalar case, this leads to restrictions for the central limit theorems, as observed by F. Oravecz.We characterize the property of independent constants from a combinatorial point of view, based on cumulants and set partitions. This characterization also holds for the operator-valued extension.Our considerations lead rather directly to very mild variations of boolean and monotone cumulants, where constants are now independent. These alternative probability theories are closely related to the usual notions. Hence, an important part of the boolean/monotone probability theories can be imported directly.We describe some standard combinatorial aspects of these variations (and their cyclic versions), such as their Möbius functions, which feature well-known combinatorial integer sequences.The new notions with independent constants seem also more strongly related to the operator-valued extension of c-free probability.

Cyclic independence: Boolean and monotone

“…The relevant partition structure turns out to be cyclic-interval partitions, which were already discussed in [7] in their search for notions of independence, similar to Boolean and monotone ones, but such that the algebra of scalars, C, is independent from any other algebra.…”

“…(1) the new notion of cyclic-Boolean independence (Sections 3) and a modification of the definition of cyclic-monotone independence given in [5] (Section 7); (2) operator models for cyclic-Boolean independence (Sections 3) and for cyclic-monotone independence (Section 7); (3) convolution formulas for the sum of independent random variables (Sections 4, 7) and their relationships to algebraic graph theory (Section 2); (4) limit theorems for sums of independent random variables (Sections 3, 7); (5) cyclic-Boolean cumulants and the relevant partition structure with cyclic-interval partitions (Section 5); (6) classification of infinitely divisible distributions for cyclic-Boolean convolution (Section 6); (7) analysis of the asymptotics of the eigenvalues of the adjacency matrices of iterated star product of graphs and iterated comb product of graphs (Sections 4, 7).…”