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Planar binary trees, noncrossing partitions and the operator-valued S-transform

Kurusch Ebrahimi-Fard, Timothe Ringeard

TL;DR

The paper investigates operator-valued free probability and the twisted multiplicativity of Voiculescu's S-transform by building a combinatorial bridge between planar binary trees and noncrossing partitions using Catalan-pair structures. It introduces boxed-convolution operations on formal multilinear function series and derives a pivotal S-transform relation that captures the interaction of products of free variables in the operator-valued setting. A key result is the operator-valued analogue of cumulant factorization for products, $k^{ab} = k^{a} \boxed{!*!}| k^{b}$, proved via a refined tree-based decomposition and a Kreweras-complement-inspired Catalan-isomorphism. Overall, the approach illuminates the interplay between combinatorial structures and operator-valued free probability, providing tools for deriving fundamental identities such as the twisted factorisation of the S-transform and related cumulant formulas.

Abstract

We revisit the twisted multiplicativity property of Voiculescu's S-transform in the operator-valued setting, using a specific bijection between planar binary trees and noncrossing partitions.

Planar binary trees, noncrossing partitions and the operator-valued S-transform

TL;DR

The paper investigates operator-valued free probability and the twisted multiplicativity of Voiculescu's S-transform by building a combinatorial bridge between planar binary trees and noncrossing partitions using Catalan-pair structures. It introduces boxed-convolution operations on formal multilinear function series and derives a pivotal S-transform relation that captures the interaction of products of free variables in the operator-valued setting. A key result is the operator-valued analogue of cumulant factorization for products, , proved via a refined tree-based decomposition and a Kreweras-complement-inspired Catalan-isomorphism. Overall, the approach illuminates the interplay between combinatorial structures and operator-valued free probability, providing tools for deriving fundamental identities such as the twisted factorisation of the S-transform and related cumulant formulas.

Abstract

We revisit the twisted multiplicativity property of Voiculescu's S-transform in the operator-valued setting, using a specific bijection between planar binary trees and noncrossing partitions.
Paper Structure (15 sections, 36 theorems, 134 equations, 1 figure)

This paper contains 15 sections, 36 theorems, 134 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Catalan pairs, binary trees and noncrossing partitions
  3. Catalan pairs
  4. Planar binary trees and noncrossing partitions
  5. The Kreweras complement as a Catalan isomorphism
  6. The S-transform of multilinear function series
  7. Series of multilinear functions
  8. The boxed convolution operations
  9. Operator-valued free probability
  10. The twisted factorisation of the S-transform
  11. The U-transform and special cases of the twisted factorisation
  12. A formula for the cumulants of a product of free variables
  13. Other bijections arising from Catalan pairs
  14. An operadic point of view
  15. Conflict of interest and data availability statements

Key Result

Lemma 2.3

Let $(C,f)$ and $(C',f')$ be Catalan pairs. Then there exists a unique isomorphism of Catalan pairs, i.e., there exists a unique bijection $\varphi: C\to C'$ such that $\varphi(1_C) = 1_{C'}$ and for all $x, y\in C$, $\varphi (f(x,y)) = f'(\varphi(x), \varphi(y))$.

Figures (1)

  • Figure 1: An example value of the function $\varphi$. Only internal vertices are shown in the tree on the left.

Theorems & Definitions (107)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 97 more