Fractional free convolution powers

Shlyakhtenko, Dimitri; Jekel, Terence Tao. With an appendix by David

doi:10.48550/arxiv.2009.01882

Cited by 8 publications

(19 citation statements)

References 27 publications

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“…If we assume that the empirical measures of eigenvalues of A N , 1 N N i=1 δ a i , converge to a limiting probability measure µ, then the (random) empirical measures of eigenvalues of A m converge to a (deterministic) measure µ τ . For integer τ this measure is the same as the free convolution of τ copies of µ, hence, µ τ can be called a fractional convolution power, see [STJ20] for a recent study and references.…”

Section: Limit Of Projectionsmentioning

confidence: 99%

Matrix addition and the Dunkl transform at high temperature

Benaych-Georges,

Cuenca,

Gorin

2021

Preprint

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We develop a framework for establishing the Law of Large Numbers for the eigenvalues in the random matrix ensembles as the size of the matrix goes to infinity simultaneously with the beta (inverse temperature) parameter going to zero. Our approach is based on the analysis of the (symmetric) Dunkl transform in this regime. As an application we obtain the LLN for the sums of random matrices as the inverse temperature goes to 0. This results in a one-parameter family of binary operations which interpolates between classical and free convolutions of the probability measures. We also introduce and study a family of deformed cumulants, which linearize this operation.

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Section: Limit Of Projectionsmentioning

confidence: 99%

Matrix addition and the Dunkl transform at high temperature

Benaych-Georges,

Cuenca,

Gorin

2021

Preprint

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“…See the recent papers [24], [12] for further discussion. The same equation also arises in free probability and random matrix theory in the context of the minor process (or the operation of fractional free convolution powers); see [23]. However, we will not use the equation (1.15) directly, preferring to work instead with its integrated form (1.14).…”

mentioning

confidence: 98%

Sendov's conjecture for sufficiently high degree polynomials

Tao¹

2020

Preprint

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Sendov's conjecture asserts that if a complex polynomial f of degree n ≥ 2 has all of its zeroes in closed unit disk {z : |z| ≤ 1}, then for each such zero λ 0 there is a zero of the derivative f in the closed unit disk {z : |z − λ 0 | ≤ 1}. This conjecture is known for n < 9, but only partial results are available for higher n. We show that there exists a constant n 0 such that Sendov's conjecture holds for n ≥ n 0 . For λ 0 away from the origin and the unit circle we can appeal to the prior work of Dégot and Chalebgwa; for λ 0 near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when λ 0 is extremely close to the unit circle); and for λ 0 near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.

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“…Shlyakhtenko [35] proved that χ increases along free convolution of µ with itself whereas Φ decreases (both suitably rescaled). Shlyakhtenko & Tao [36] showed monotonicity along the entire flow µ k for real k ≥ 1. Conversely, on the side of polynomials, we showed [39] that…”

mentioning

confidence: 99%

“…We first clarify the meaning of 'formally'. In a recent paper, Shlyakhtenko & Tao [36] derived, formally, a PDE for the evolution of the µ k . This PDE happens to be the same PDE (expressed in a different coordinate system) that was formally derived by the author for the evolution of u(t, x) [38].…”

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confidence: 99%

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Free Convolution Powers via Roots of Polynomials

Steinerberger¹

2020

Preprint

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Let µ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power µ k for any real k ≥ 1. The purpose of this short note is to give an elementary description of µ k in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.Given G µ , we define the R−transform R µ (s) for sufficiently small complex s by demanding that 1for all sufficiently large z. The free convolution µ ν is then the unique compactly supported measure for whichfor all sufficiently small s. A fundamental result due to Voiculescu is the free central limit theorem: if µ is a compactly supported probability measure with mean 0 and variance 1, then suitably rescaled copies of µ k converge to the semicircular distribution. This notion can be extended to the reals.Theorem (Fractional Free Convolution Powers exist, [5,24]). Let µ be a compactly supported probability measure on R and assume k ≥ 1 is real. Then there exists a unique compactly supported probability measure µ k such thatfor all s sufficiently small.

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Fractional free convolution powers

Cited by 8 publications

References 27 publications

Matrix addition and the Dunkl transform at high temperature

Matrix addition and the Dunkl transform at high temperature

Sendov's conjecture for sufficiently high degree polynomials

Free Convolution Powers via Roots of Polynomials

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