Equidistribution of zeros of random polynomials

Pritsker, Igor E.; Ramachandran, Koushik

doi:10.1016/j.jat.2016.12.001

Cited by 15 publications

(22 citation statements)

References 26 publications

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“…As a corollary of Theorem 5.3, we resolve a conjecture of Pristker and Ramachandran ( [20], Conjecture 2.5). Very roughly, they asked if there exist i.i.d.…”

Section: Introductionsupporting

confidence: 56%

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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“…As a corollary of Theorem 5.3, we resolve a conjecture of Pristker and Ramachandran ( [20], Conjecture 2.5). Very roughly, they asked if there exist i.i.d.…”

Section: Introductionsupporting

confidence: 56%

Asymptotic zero distribution of random orthogonal polynomials

Bloom

Dauvergne²

2019

Ann. Probab.

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“…Similarly, ρ 0,1 is called a density of complex zeros being an expectation of the empirical measure µ counting non-real zeros. Its limit behaviour as n → ∞ is of a great interest as well, see [31], [17], [16], [19], [29], [30], and the references given there.…”

Section: Zeros Of Random Polynomialsmentioning

confidence: 99%

Joint distribution of conjugate algebraic numbers: A random polynomial approach

Götze

Koleda

Zaporozhets

2020

Advances in Mathematics

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“…This work is a sequel to [8] where we showed that zeros of a sequence of random polynomials {P n } n (spanned by an appropriate basis) associated to a Jordan domain G with analytic boundary L, equidistribute near L, i.e. distribute according to the equilibrium measure of L. We refer the reader to [8] for references to the literature on random polynomials. In this note, we extend the above result to Jordan domains with lesser regularity, namely domains with C 2,α boundary, see Theorem 1.1 below.…”

Section: Introductionmentioning

confidence: 99%

Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains

Pritsker

Ramachandran

2019

Comput. Methods Funct. Theory

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We consider the zero distribution of random polynomials of the form P n (z) = n k=0 a k B k (z), where {a k } ∞ k=0 are non-trivial i.i.d. complex random variables with mean 0 and finite variance. Polynomials {B k } ∞ k=0 are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain G whose boundary is C 2,α smooth. We show that the zero counting measures of P n converge almost surely to the equilibrium measure on the boundary of G. We also show that if {a k } ∞ k=0 are i.i.d. random variables, and the domain G has analytic boundary, then for a random series of the form f (z) = ∞ k=0 a k B k (z), ∂G is almost surely a natural boundary for f (z).

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Equidistribution of zeros of random polynomials

Cited by 15 publications

References 26 publications

Asymptotic zero distribution of random orthogonal polynomials

Asymptotic zero distribution of random orthogonal polynomials

Joint distribution of conjugate algebraic numbers: A random polynomial approach

Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains

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