Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature

Flasche, Hendrik; Kabluchko, Zakhar

doi:10.1007/s10959-018-0843-z

Cited by 6 publications

(2 citation statements)

References 38 publications

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“…Various authors subsequently worked on this problem, leading to significant developments during 1940s-1970s, with seminal contributions of Kac [17], Littlewood and Offord [19,20,21], Ibragimov and Maslova [12,14,15,13,22,23], among others. Recently, there has been a renewed interest in this problem [6,16,11,2,9,31,32,5,8,27], in particular Tao and Vu [33] developed a new framework to study the real roots of random polynomials, adapting their methods from random matrix theory. See also [25,3,4,26] for some further development of the methods in [33].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications

Q.¹

2021

Electron. J. Probab.

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This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of the coefficients have power growth and count the average number of real zeros. Almost all previous results require coefficients with zero mean, and it is nontrivial to extend these results to the general case. Our approach is based on a novel comparison principle that reduces the general situation to the mean-zero setting. As applications, we obtain new results for the Kac polynomials, hyperbolic random polynomials, their derivatives, and generalizations of these polynomials. The proof features new logarithmic integrability estimates for random polynomials (both local and global) and fairly sharp estimates for the local number of real zeros.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications

Q.¹

2021

Electron. J. Probab.

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“…In parallel, Flasche and Kabluchko [12,13] studied real roots of random series with regularly varying coefficients…”

Section: Introductionmentioning

confidence: 99%

The number of limit cycles bifurcating from a randomly perturbed center

Lundberg¹,

Nguyen²

2021

Preprint

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We consider the average number of limit cycles that bifurcate from a randomly perturbed linear center where the perturbation consists of random (bivariate) polynomials with independent coefficients. This problem reduces, by way of classical perturbation theory of the Poincaré first return map, to a problem on the real zeros of a random univariate poly-c m ξ m x m with independent coefficients ξ m having mean zero, variance 1 and c m ∼ m −1/2 . This polynomial belongs to the class of generalized Kac polynomials, but it falls right at the edge of the family of well-studied cases. We provide asymptotic estimates for its average number of positive real zeros, thus addressing the question on bifurcating limit cycles.

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On the Zeros of Non-Analytic Random Periodic Signals

Angst

Poly

2020

International Mathematics Research Notices

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In this paper, we investigate the local universality of the number of zeros of a random periodic signal of the form $S_n(t)=\sum _{k=1}^n a_k f(k t)$, where $f$ is a $2\pi -$periodic function satisfying weak regularity conditions and where the coefficients $a_k$ are i.i.d. random variables, which are centered with unit variance. In particular, our results hold for continuous piecewise linear functions. We prove that the number of zeros of $S_n(t)$ in a shrinking interval of size $1/n$ converges in law as $n$ goes to infinity to the number of zeros of a Gaussian process whose explicit covariance only depends on the function $f$ and not on the common law of the random coefficients $(a_k)$. As a byproduct, this entails that the point measure of the zeros of $S_n(t)$ converges in law to an explicit limit on the space of locally finite point measures on $\mathbb R$ endowed with the vague topology. The standard tools involving the regularity or even the analyticity of $f$ to establish such kind of universality results are here replaced by some high-dimensional Berry–Esseen bounds recently obtained in [ 7]. The latter allow us to prove functional Central Limit Theorems in $C^1$ or Lipschitz topology in situations where usual criteria cannot be applied due to the lack of regularity.

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Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature

Abstract: Let ξ0, ξ1, .

Cited by 6 publications

References 38 publications

Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications

Real roots of random polynomials with coefficients of polynomial growth: a comparison principle and applications

The number of limit cycles bifurcating from a randomly perturbed center

On the Zeros of Non-Analytic Random Periodic Signals

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