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Infinitesimal moments in free and c-free probability and Motzkin paths

Romuald Lenczewski

TL;DR

The paper develops a moment-based, tensor-product framework for infinitesimal free and infinitesimal conditional freeness, recasting first-order corrections via continuous deformations of Motzkin-functionals indexed by Motzkin words. A key finding is that the first derivative contributions are highly selective: pyramid Motzkin paths (one local maximum) govern infinitesimal freeness, while concatenations of a pyramid and a flat path govern infinitesimal c-freeness, yielding Leibniz-type formulas for $\varphi'$-moments and their c-free analogues. The approach relies on level return partitions and Boolean extensions to decompose moments into products of cumulants along blocks determined by the Motzkin structure, with higher-order derivatives controlled by the number of local maxima. Overall, the work provides a unified, combinatorially explicit method to compute infinitesimal corrections in free and conditional free contexts, relating infinitesimality to Motzkin-path geometry and tensor-product constructions. This offers new tools for understanding finite-size effects in random matrix models and deepens the connection between noncommutative independence and Motzkin-path combinatorics.

Abstract

Infinitesimal moments associated with infinitesimal freeness and infinitesimal conditional freeness are studied. For free random variables, we consider continuous deformations of moment functionals associated with Motzkin paths $w$, which provide a decomposition of their moments, and we compute their derivatives at zero. We show that the first-order derivative of each functional vanishes unless the path has exactly one local maximum. Geometrically, this means that $w$ is a pyramid path, which is consistent with the characteristic formula for alternating moments of infinitesimally free centered random variables. In this framework, infinitesimal Boolean independence is also obtained and it corresponds to flat paths. A similar approach is developed for infinitesimal conditional freeness, for which we show that the only moment functionals that have a non-zero first-order derivative are associated with concatenations of a pyramid path and a flat path. This charaterization leads to a Leibniz-type definition of infinitesimal conditional freeness at the level of moments.

Infinitesimal moments in free and c-free probability and Motzkin paths

TL;DR

The paper develops a moment-based, tensor-product framework for infinitesimal free and infinitesimal conditional freeness, recasting first-order corrections via continuous deformations of Motzkin-functionals indexed by Motzkin words. A key finding is that the first derivative contributions are highly selective: pyramid Motzkin paths (one local maximum) govern infinitesimal freeness, while concatenations of a pyramid and a flat path govern infinitesimal c-freeness, yielding Leibniz-type formulas for -moments and their c-free analogues. The approach relies on level return partitions and Boolean extensions to decompose moments into products of cumulants along blocks determined by the Motzkin structure, with higher-order derivatives controlled by the number of local maxima. Overall, the work provides a unified, combinatorially explicit method to compute infinitesimal corrections in free and conditional free contexts, relating infinitesimality to Motzkin-path geometry and tensor-product constructions. This offers new tools for understanding finite-size effects in random matrix models and deepens the connection between noncommutative independence and Motzkin-path combinatorics.

Abstract

Infinitesimal moments associated with infinitesimal freeness and infinitesimal conditional freeness are studied. For free random variables, we consider continuous deformations of moment functionals associated with Motzkin paths , which provide a decomposition of their moments, and we compute their derivatives at zero. We show that the first-order derivative of each functional vanishes unless the path has exactly one local maximum. Geometrically, this means that is a pyramid path, which is consistent with the characteristic formula for alternating moments of infinitesimally free centered random variables. In this framework, infinitesimal Boolean independence is also obtained and it corresponds to flat paths. A similar approach is developed for infinitesimal conditional freeness, for which we show that the only moment functionals that have a non-zero first-order derivative are associated with concatenations of a pyramid path and a flat path. This charaterization leads to a Leibniz-type definition of infinitesimal conditional freeness at the level of moments.
Paper Structure (9 sections, 27 theorems, 140 equations, 4 figures)

This paper contains 9 sections, 27 theorems, 140 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Infinitesimal freeness and infinitesimal idempotents
  3. Motzkin paths and level return partitions
  4. Infinitesimal deformations of Motzkin functionals
  5. Partial tensor evaluations
  6. Formula for moments
  7. Infinitesimal Motzkin functionals
  8. Higher order derivatives
  9. Infinitesimal conditional freeness

Key Result

Proposition 2.1

Let $({\mathcal{A}}, \varphi, \varphi')$ be an infinitesimal noncommutative probability space and let $\{{\mathcal{A}}_{i}:\,i\in I\}$ be a family of unital subalgebras of ${\mathcal{A}}$. This family is infinitesimally free with respect to $(\varphi, \varphi')$ if and only if

Figures (4)

  • Figure 1: Three Motzkin paths and the associated level return partitions. Local maxima of paths and the corresponding singletons in level return partitions are marked with hollow circles.
  • Figure 2: Pyramid Motzkin path of length $10$ and the corresponding fully nested level return partition with a central singleton block.
  • Figure 3: Motzkin paths of length $5$ with two local maxima.
  • Figure 4: Motzkin paths of length $4$ contributing to infinitesimal c-freeness. Local maxima of special type are marked with hollow circles.

Theorems & Definitions (58)

  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 48 more