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Combinatorics of cyclic-conditional freeness

Octavio Arizmendi, Guillaume Cébron, Nicolas Gilliers

Abstract

This work investigates the combinatorial structures underlying cyclic conditional freeness and introduces cumulants that serve to linearize the cyclic conditional additive convolution. In the process, we establish the notion of "cyclic freeness", demonstrating its equivalence to infinitesimal freeness in the presence of tracial states. Furthermore, we show that cyclic conditional freeness can be reduced to cyclic freeness through a multivariate extension of the inverse Markov-Krein transform.

Combinatorics of cyclic-conditional freeness

Abstract

This work investigates the combinatorial structures underlying cyclic conditional freeness and introduces cumulants that serve to linearize the cyclic conditional additive convolution. In the process, we establish the notion of "cyclic freeness", demonstrating its equivalence to infinitesimal freeness in the presence of tracial states. Furthermore, we show that cyclic conditional freeness can be reduced to cyclic freeness through a multivariate extension of the inverse Markov-Krein transform.
Paper Structure (22 sections, 39 theorems, 233 equations, 3 figures)

This paper contains 22 sections, 39 theorems, 233 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Organization of the paper
  4. Cyclic and infinitesimal freeness
  5. Background and notation for partitions
  6. Cumulants for cyclic freeness
  7. Proof of Theorem \ref{['th:cyclicfree']}
  8. From conditional freeness to cyclic and infinitesimal freeness
  9. Cyclically alternating product of cyclically free variables
  10. Cyclic conditional freeness
  11. Proof of Theorem \ref{['th:fromfreetocyclic']}
  12. Proof of Theorem \ref{['th:vanishing_of_mixed_cumulants']}
  13. Cumulants for cyclic Boolean independence
  14. Convolutions and limit theorems
  15. Cyclic conditional additive convolution
  16. ...and 7 more sections

Key Result

Proposition 1.4

The multivariate inverse Markov-Krein transform of ${\varphi}$, defined by eqn:MKtransform is equal to the linear form $[{\varphi}]$.

Figures (3)

  • Figure 1: Each red arrows means that we use the multivariate Markov-Krein transform (we call those arrows cyclic companions in this text). Each black arrow means the target independence is obtained under further assumptions on the linear functionals and the independence at the source. The grey arrow represents an extension of a result by Février, Mastnak, Nica and Szpojankowski. The blue arrows proceed from considerations as in Proposition 2.16 in cebron2022freeness stating the connection between the monotone and cyclic monotone independences.
  • Figure 2: Let $\pi=\{ \{ \{1\}, \{2,6,15\}, \{3,5\},\{4\},\{7,11,12\}, \{8,10\},\{9\} \{13,14\},\{16\} \}\}$. On the left side, the block $V$ is drawn with dots, and the corresponding chains of the preorder $\prec_{V}$ are represented by arrows. On the right side the block $R$ in the Kreweras complement of $\pi$ is filled with black, $R=\{\{5',7'\},\{15'\}\}$. The chains of the preorder $\prec_{R}$ are also represented by arrows.
  • Figure 3: The graphs $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$ and $G_1*G_2$ together with their branches $\Gamma_1$ and $\Gamma_2$.

Theorems & Definitions (89)

  • Definition 1.1: Cyclic freeness
  • Example 1.2
  • Remark 1.3
  • Proposition 1.4
  • Remark 1.5
  • proof
  • Definition 1.6: Conditional freeness bozejko1996convolution
  • Definition 1.7: Cyclic conditional freeness cebron2022asymptotic
  • Remark 1.8
  • Theorem 1.9
  • ...and 79 more