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

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.

Summary

This paper introduces a novel combinatorial approach using Motzkin paths to analyze infinitesimal moments in free and c-free probability. By considering continuous deformations of moment functionals associated with these paths, the authors uncover deep connections between infinitesimal freeness and the geometric properties of the paths, offering a new perspective on first-order corrections to asymptotic freeness.

Key Insights

• •The paper establishes a framework for decomposing moments using Motzkin paths and multilinear moment functionals, revealing how these functionals relate to the moments of free random variables.
• •A key finding is the characterization of pyramid Motzkin paths as uniquely linked to the Leibniz-type definition of infinitesimal freeness, demonstrating that only these paths have non-vanishing first-order derivatives under specific conditions.
• •The authors demonstrate that infinitesimal Boolean independence corresponds to 'flat' Motzkin paths, leading to a Boolean analog of the Leibniz-type formula and expanding the applicability of the framework.
• •The work generalizes to infinitesimal conditional freeness, deriving a Leibniz-type definition for infinitesimal moments using concatenated Motzkin paths, providing a c-free analog to existing results.

Practical Implications

• •The combinatorial framework developed in this paper can be applied to analyze first-order corrections to asymptotic freeness in random matrix models, providing a powerful tool for understanding the behavior of large random matrices.
• •Future research can explore higher-order infinitesimal freeness using the geometric properties of Motzkin paths, potentially leading to a more complete understanding of the structure of free probability.
• •Extending the results to the general case of infinitesimal monotone independence represents another avenue for future research, broadening the scope of the framework.
• •The connection to idempotents and Boolean extensions suggests potential applications in tensor product constructions and other areas of operator algebra.

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