Cumulants for finite free convolution

Arizmendi, Octavio; Perales, Daniel

doi:10.1016/j.jcta.2017.11.012

Cited by 14 publications

(25 citation statements)

References 18 publications

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“…Moreover, when ( ) = ( − 1) , the second equation in Theorem 1.2 recovers the rst-order asymptotics of proven in [AP18], namely…”

Section: Introductionmentioning

confidence: 54%

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Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions

Arizmendi¹,

Garza-Vargas²,

Perales³

2021

Preprint

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Given two polynomials ( ), ( ) of degree , we give a combinatorial formula for the nite free cumulants of ( )⊠ ( ). We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of di erent genera.This topological expansion, on the one hand, deepens the connection between the theories of nite free probability and free probability, and in particular proves that ⊠ converges to ⊠ as goes to in nity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the in nitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of in nitesimal distributions for real random matrices.Finally, building o our results we give a new short and conceptual proof of a recent result [Ste20, HK20] that connects root distributions of polynomial derivatives with free fractional convolution powers. * O.A. gratefully acknowledges nancial support by the grants Conacyt A1-S-9764 and SFB TRR 195.

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“…Moreover, when ( ) = ( − 1) , the second equation in Theorem 1.2 recovers the rst-order asymptotics of proven in [AP18], namely…”

Section: Introductionmentioning

confidence: 54%

“…Of greatest relevance to this paper is the combinatorial approach, based on cumulants, for the nite free additive convolution that was introduced by Arizmendi and Perales [AP18]. These nite free cumulants approach free cumulants as goes to in nity and share many of their properties.…”

Section: Introductionmentioning

confidence: 99%

Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions

Arizmendi¹,

Garza-Vargas²,

Perales³

2021

Preprint

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“…Following the definitions in this paper, [10,11] have shown that by taking appropriate limits our finite free convolutions yield the standard free convolutions in free probability theory. Thus, expected characteristic polynomials provide an alternative "discretization" of free convolutions from the typical one involving random matrices.…”

Section: Motivation and Related Resultsmentioning

confidence: 99%

Finite free convolutions of polynomials

Marcus

Spielman

Srivastava

2022

Probab. Theory Relat. Fields

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We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szegö in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.

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“…The connection was formalized in [12], where it was shown that the inequalities derived for two of the convolutions studied in [13] -the symmetric additive and multiplicative convolutionsconverge to the R-and S-transform identities of free probability (respectively). Since the release of [12], a number of advances have been made in understanding the relationship between free probability and polynomial convolutions, most notably the work of Arizmendi and Perales [1] in developing a combinatorial framework for finite free probability using finite free cumulants (the approach in [12] is primarily analytic).…”

Section: Introductionmentioning

confidence: 99%

A rectangular additive convolution for polynomials

Gribinski¹,

Marcus²

2019

Preprint

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We define the rectangular additive convolution of polynomials with nonnegative real roots as a generalization of the asymmetric additive convolution introduced in [13]. We then prove a sliding bound on the largest root of a convolution that extends the one proved in [13]. The main tool used in the analysis is a differential operator derived from the "rectangular Cauchy transform" introduced in [3]. The proof is inductive, with the base case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer polynomials which may be of independent interest.

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Cumulants for finite free convolution

Abstract: In this paper we define cumulants for finite free convolution. We give a moment-cumulant formula and show that these cumulants satisfy desired properties: they are additive with respect to finite free convolution and they approach free cumulants as the dimension goes to infinity.

Cited by 14 publications

References 18 publications

Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions

Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions

Finite free convolutions of polynomials

A rectangular additive convolution for polynomials

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