Combinatorics of NC-probability spaces with independent constants

Abstract

The boolean and monotone notions of independence lack the property of independent constants. We address this problem from a combinatorial point of view (based on cumulants defined from weights on set-partitions, in the general framework of operator-valued probability spaces). We show that if the weights are singleton inductive (SI), then all higher-order cumulants involving constants vanish, just as in the free and classical case. Our combinatorial considerations lead rather directly to mild variations of boolean and monotone probability theories which are closely related to the usual notions. The SI-boolean case is related to c-free and Fermi convolutions. We also describe some standard combinatorial aspects of the SI-boolean and cyclic-boolean lattices, such as their Möbius functions, featuring well-known combinatorial integer sequences.

Cyclic independence: Boolean and monotone

“…Cyclic-Boolean independence gives rise to an exchangeability system and we can define and compute the (multivariate) cyclic-Boolean cumulants using the methods of [13,11]. 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.…”

“…(5) cyclic-Boolean cumulants and the relevant partition structure with cyclicinterval 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). Recently Collins, Leid and Sakuma found a different matrix model for monotone independence and cyclic monotone independence [6].…”

Towards a classification of multi-faced independences: a combinatorial approach

“…In [DAGSV22], Diaz-Aguilera, Gaxiola, Santos, and Vargas characterize when the moment cumulant relation associated with weights on partitions leads to independent constants, finding this to be the case if and only if the weights do not change when removing or inserting a singleton from or to the partition. Manzel and Schürmann discuss in [MS17, Rem.…”