Free Convolution Powers via Roots of Polynomials

Steinerberger, Stefan

doi:10.48550/arxiv.2009.03869

Cited by 5 publications

(6 citation statements)

References 36 publications

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“…Secondly, we derive the interesting relation between derivatives of a polynomial and free additive convolution powers, as observed by Steinerberger [Ste20] and proved by Hoskins and Kobluchko [HK20]. The main observation here, which we prove in Section 3.3, is that when…”

Section: Introductionmentioning

confidence: 54%

“…After xing ∈ (0, 1) we will be interested in the asymptotic behavior, as → ∞, of the root distributions of ⌊(1− ) ⌋ ( ), where denotes di erentiation with respect to . Using the PDE characterization of free fractional convolution powers obtained by Tao and Shlyakhtenko [ST20], in [Ste19] and [Ste20] Steinerberger informally showed that, under the above setup, the empirical root distributions (after proper normalization) of the polynomials ⌊(1− ) ⌋ ( ) converge to ⊞1/ as → ∞. This was later formally proven by Hoskins and Kobluchko [HK20] by directly calculating the asymptotics of the -transform of the derivatives of .…”

Section: Derivatives Of Polynomials Tend To Free Fractional Powersmentioning

confidence: 92%

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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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Section: Introductionmentioning

confidence: 54%

Section: Derivatives Of Polynomials Tend To Free Fractional Powersmentioning

confidence: 92%

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

Arizmendi¹,

Garza-Vargas²,

Perales³

2021

Preprint

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“…Besides its aesthetic aspect, this equation has many interesting features. Shlyakhtenko and Tao [36] derived the same equation in the context of free probability and random matrix theory (see also [39]). However, our motivation comes from the links between this equation and many models studied in fluid dynamics.…”

Section: Introductionmentioning

confidence: 95%

On the dynamics of the roots of polynomials under differentiation

Alazard¹,

Lazar²,

Nguyen³

2021

Preprint

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This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and Terence Tao on free convolution. Rafael Granero-Belinchón obtained a global well-posedness result for positive initial data small enough in a Wiener space, and recently Alexander Kiselev and Changhui Tan proved a global well-posedness result for any positive initial data in the Sobolev space H s (S) with s > 3/2. In this paper, we consider the Cauchy problem in the critical space H 1/2 (S). Two interesting new features, at this level of regularity, are that the equation can be written in the form ∂tu + V ∂xu + γΛu = 0, where γ is non-negative but not bounded from below and V / √ γ is not bounded. Therefore, the equation is only weakly parabolic. We prove that nevertheless the Cauchy problem is well posed locally in time and that the solutions are smooth for positive times. Combining this with the results of Kiselev and Tan, this gives a global well-posedness result for any positive initial data in H 1/2 (S). Our proof relies on sharp commutator estimates and introduces a strategy to prove a local well-posedness result in a situation where the lifespan depends on the profile of the initial data and not only on its norm.

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“…While in [20], the more "explicit" strategy of proof relies on complex analysis consequences of the (proved) fact that the roots of the iterated derivatives are distributed according to the free multiplicative convolution of µ and a free unitary Poisson distribution. This last work can be related to [30], [35], [18] and [2] for free probabilities and also to [3] for the saddle point technique of proof.…”

Section: Introductionmentioning

confidence: 99%

Modeling complex root motion of real random polynomials under differentiation

Galligo

2022

Preprint

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In this paper, we consider nonlocal, nonlinear partial differential equations to model anisotropic dynamics of complex root sets of random polynomials under differentiation. These equations aim to generalise the recent PDE obtained by Stefan Steinerberger (2019) in the real case, and the PDE obtained by Sean O'Rourke and Stefan Steinerberger (2020) in the radial case, which amounts to work in 1D. These PDEs approximate dynamics of the complex roots for random polynomials of sufficiently high degree n. The unit of the time t corresponds to n differentiations, and the increment ∆t corresponds to 1 n . The general situation in 2D, in particular for complex roots of real polynomials, was not yet addressed. The purpose of this paper is to present a first attempt in that direction. We assume that the roots are distributed according to a regular distribution with a local homogeneity property (defined in the text), and that this property is maintained under differentiation. This allows us to derive a system of two coupled equations to model the motion. Our system could be interesting for other applications. The paper is illustrated with examples computed with the Maple system.

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Free Convolution Powers via Roots of Polynomials

Cited by 5 publications

References 36 publications

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

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

On the dynamics of the roots of polynomials under differentiation

Modeling complex root motion of real random polynomials under differentiation

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