Equidistribution of zeros of random holomorphic sections

Bayraktar, Turgay

doi:10.1512/iumj.2016.65.5910

Cited by 36 publications

(59 citation statements)

References 43 publications

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“…coefficients so that we still obtain the same limiting behaviour of the zeros. In [1], Bayraktar established the almost sure weak* convergence of the zeros of random polynomials of the form (2) for i.i.d. coefficients ξ i with a continuous Lebesgue density, satisfying P(|ξ 0 | ≥ e |z| ) = O(|z| −ρ ), where polynomials are considered in d variables and ρ > d + 1.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic zero distribution of random orthogonal polynomials

Bloom

Dauvergne²

2019

Ann. Probab.

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We consider random polynomials of the form H n (z) = n j=0 ξ j q j (z) where the {ξ j } are i.i.d non-degenerate complex random variables, and the {q j (z)} are orthonormal polynomials with respect to a compactly supported measure τ satisfying the Bernstein-Markov property on a regular compact set K ⊂ C. We show that if P(|ξ 0 | > e |z| ) = o(|z| −1 ), then the normalized counting measure of the zeros of H n converges weakly in probability to the equilibrium measure of K. This is the best possible result, in the sense that the roots of G n (z) = n j=0 ξ j z j fail to converge in probability to the appropriate equilibrium measure when the above condition on the ξ j is not satisfied. In addition, we give a multivariable version of this result.We also consider random polynomials of the form n k=0 ξ k f n,k z k , where the coefficients f n,k are complex constants satisfying certain conditions, and the random variables {ξ k } satisfy E log(1 + |ξ 0 |) < ∞. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.

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Section: Introductionmentioning

confidence: 99%

Asymptotic zero distribution of random orthogonal polynomials

Bloom

Dauvergne²

2019

Ann. Probab.

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“…In this work, we employ methods of pluripotential theory (cf. [SZ04,DS06a,BS07,CM15,BL15,Bay16]) which is extensively used in the dynamical study of holomorphic maps (see [FS95] and references therein). Along the way, we develop a pluripotential theory for plurisubharmonic (psh for short) functions which are dominated by the support function of P (up to a constant) in logarithmic coordinates on (C * ) m .…”

Section: Introductionmentioning

confidence: 99%

Zero distribution of random sparse polynomials

Bayraktar¹

2017

Michigan Math. J.

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Abstract. We study asymptotic zero distribution of random Laurent polynomials whose support are contained in dilates of a fixed integral polytope P as their degree grow. We consider a large class of probability distributions including the ones induced from i.i.d. random coefficients whose distribution law has bounded density with logarithmically decaying tails as well as moderate measures defined over the projectivized space of Laurent polynomials. We obtain a quantitative localized version of Bernstein-Kouchnirenko Theorem.

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“…Asymptotics for the density function ρ n (x) in the case when the random variables {η k } are i.i.d. standard Gaussian has been well studied when the spanning functions are trigonometric functions [36], polynomials orthogonal on the real line ( [6], [7], [2], [28], [29], [31], [39]), and polynomials orthogonal on the unit circle ( [39], [1], [38]). As an application we consider the case…”

Section: Introductionmentioning

confidence: 99%

Real zeros of random sums with i.i.d. coefficients

Yeager

2020

Colloq. Math.

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Let {f k } be a sequence of entire functions that are real valued on the real-line. We study the expected number of real zeros of random sums of the form P n (z) = n k=0 η k f k (z), where {η k } are real valued i.i.d. random variables. We establish a formula for the density function ρ n for the expected number of real zeros of P n . As a corollary, taking the random variables {η k } to be i.i.d. standard Gaussian, appealing to Fourier inversion we recover the representation for the density function previously given by Vanderbei through means of a different proof. Placing the restrictions on the common characteristic function φ of {η k } that |φ(s)| ≤ (1 + as 2 ) −q , with a > 0 and q ≥ 1, as well as that φ is three times differentiable with each the second and third derivatives being uniformly bounded, we achieve an upper bound on the density function ρ n with explicit constants that depend only on the restrictions on φ. As an application we considered the limiting value of ρ n when the spanning functions f k (z) = p k (z), k = 0, 1, . . . , n, where {p k } are Bergman polynomials on the unit disk.

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Equidistribution of zeros of random holomorphic sections

Cited by 36 publications

References 43 publications

Asymptotic zero distribution of random orthogonal polynomials

Asymptotic zero distribution of random orthogonal polynomials

Zero distribution of random sparse polynomials

Real zeros of random sums with i.i.d. coefficients

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