Cumulant–Cumulant Relations in Free Probability Theory from Magnus’ Expansion

Celestino, Adrian; Ebrahimi‐Fard, Kurusch; Patras, Frédéric; Perales, Daniel

doi:10.1007/s10208-021-09512-0

Cited by 10 publications

(21 citation statements)

References 26 publications

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“…In our [6], the same combinatorics of trees and the same combinatorial coefficients as in [20] appeared. The reason for this appearance was easy to guess: in non-commutative probability, a pre-Lie structure underlies our formulas, whereas trees encode the free pre-Lie algebra on one generator.…”

Section: Introductionmentioning

confidence: 58%

“…The following two theorems were obtained in [6] by direct computations, we show here how they can be deduced from pre-Lie forest formulas. Theorem 9.9.…”

Section: Theorem 94 ([2]mentioning

confidence: 91%

“…We now introduce an important example of pre-Lie algebras for our purposes. Its introduction is motivated by the study of cumulant-cumulant relations in non-commutative probability: see our [6] and the forthcoming developments in this article. Henceforth, let X be a finite alphabet and denote by X * the set of non-empty words over X.…”

Section: A Pre-lie Algebra Of Wordsmentioning

confidence: 99%

“…Recall that the Baker-Campbell-Hausdorff formula computes the logarithm of the fundamental solution of such an equation. The coefficients associated to a rooted tree defined in this section have appeared in Murua's analysis of the continuous Baker-Campbell-Hausdorff problem in a Hall basis ( [20]), and, more recently, as the coefficients defining the cumulant-moment and cumulant-cumulant formulas associated to monotone independence in the context of non-commutative probability theory ( [6]). We feature once again that one of our aims was to deepen the understanding of Murua's article and its relations to our work in non-commutative probability: the methods we develop hereafter shed a new light on his formulas and constructions on (non-planar) trees by showing that they emerge naturally from our forest formulas.…”

Section: Murua's Coefficients and The Magnus Operator Imentioning

confidence: 99%

“…The present article was motivated by our recent work on cumulant-cumulant relations in noncommutative probability [6] and the work of Murua [20]. Murua's work is a key contribution to one of the most classical problems: the Baker-Campbell-Hausdorff (BCH) problem, that is the computation of the logarithm of the solution of a non-autonomous matrix-valued linear differential equation.…”

Section: Introductionmentioning

confidence: 99%

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A forest formula for pre-Lie exponentials, Magnus' operator and cumulant-cumulant relations

Celestino¹,

Patras²

2022

Preprint

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Forest formulas that generalize Zimmermann's forest formula in quantum field theory have been obtained for the computation of the antipode in the dual of enveloping algebras of pre-Lie algebras. In this work, largely motivated by Murua's analysis of the Baker-Campbell-Hausdorff formula, we show that the same ideas and techniques generalize and provide effective tools to handle computations in these algebras, which are of utmost importance in numerical analysis and related areas. We illustrate our results by studying the action of the pre-Lie exponential and the Magnus operator in the free pre-Lie algebra and in a pre-Lie algebra of words originating in free probability. The latter example provides combinatorial formulas relating the different brands of cumulants in non-commutative probability.

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Section: Introductionmentioning

confidence: 58%

“…The following two theorems were obtained in [6] by direct computations, we show here how they can be deduced from pre-Lie forest formulas. Theorem 9.9.…”

Section: Theorem 94 ([2]mentioning

confidence: 91%

Section: A Pre-lie Algebra Of Wordsmentioning

confidence: 99%

Section: Murua's Coefficients and The Magnus Operator Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A forest formula for pre-Lie exponentials, Magnus' operator and cumulant-cumulant relations

Celestino¹,

Patras²

2022

Preprint

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Algebraic Structures Underlying Quantum Independences: Theory and Applications

Chétrite¹,

Patras²

2023

Ann. Henri Poincaré

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Fertilitopes

Defant

2023

Discrete Comput Geom

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We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s. Associated to each permutation $$\pi $$ π is a particular set $$\mathcal V(\pi )$$ V ( π ) of integer compositions that appears in a formula for the fertility of $$\pi $$ π , which is defined to be $$|s^{-1}(\pi )|$$ | s - 1 ( π ) | . These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that $$\mathcal V(\pi )$$ V ( π ) is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of $$\pi $$ π to be the convex hull of $$\mathcal V(\pi )$$ V ( π ) , and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of $$\pi $$ π directly from $$\pi $$ π using Bousquet-Mélou’s notion of the canonical tree of $$\pi $$ π . As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that $$\sum _{\sigma \in s^{-1}(\pi )}x^{\textrm{des}(\sigma )+1}$$ ∑ σ ∈ s - 1 ( π ) x des ( σ ) + 1 is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

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Cumulant–Cumulant Relations in Free Probability Theory from Magnus’ Expansion

Cited by 10 publications

References 26 publications

A forest formula for pre-Lie exponentials, Magnus' operator and cumulant-cumulant relations

A forest formula for pre-Lie exponentials, Magnus' operator and cumulant-cumulant relations

Algebraic Structures Underlying Quantum Independences: Theory and Applications

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