arXiv subjects

Yen Q. Do

Publications and source records attributed to Yen Q. Do.

5 recordsLinked to original sources

arXiv2026

Real roots of non-centered random polynomials

We study the fluctuations of the number of real roots of random polynomials with independent, nonzero-mean coefficients. Such non-centered ensembles arise naturally in signal-plus-noise models and in random perturbations of deterministic polynomials. While Ibragimov and Maslova (1971) established the leading asymptotics of the expected number of real roots for non-centered polynomials with i.i.d. coefficients, the corresponding variance asymptotics and central limit theorem have remained open for more than fifty years. This stands in sharp contrast to the centered case, where the fluctuation theory is now well understood across a wide range of ensembles. We resolve this gap by developing novel comparison principles that reduce the fluctuation theory of a non-centered ensemble to that of its centered counterpart. These principles yield sharp variance asymptotics and central limit theorems for broad classes of ensembles, including Kac and hyperbolic polynomials, their derivatives, and related extensions. In particular, for both Kac and hyperbolic polynomials, the leading variance constant equals exactly one-half of that in the centered case, reflecting asymmetric suppression of fluctuations across the two regions where roots concentrate. Our results provide the first comprehensive fluctuation theory for the number of real roots of non-centered random polynomials.

math.PR↗

arXiv2025

Real roots of random polynomials: asymptotics of the variance

We compute the precise leading asymptotics of the variance of the number of real roots for a large class of random polynomials, where the random coefficients have polynomial growth. Our results apply to many classical ensembles, including the Kac polynomials, hyperbolic polynomials, their derivatives, and any linear combinations of these polynomials. Prior to this paper, such asymptotics were established only for the Kac polynomials in the 1970s, with the seminal contribution of Maslova. The main ingredients of the proof are new asymptotic estimates for the two-point correlation function of the real roots, revealing geometric structures in the distribution of the real roots of these random polynomials. As a corollary, we obtain asymptotic normality for the real roots of these random polynomials, extending and strengthening a related result of O. Nguyen and V. Vu.

math.PR↗

arXiv2024

A strong law of large numbers for real roots of random polynomials

We consider random polynomials $p_n(x)=ξ_0+ξ_1+\dots+ξ_n x^n$ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded $(2+ε)^{th}$ moment (for some $ε>0$), also known as the Kac polynomials. Let $N_n$ denote the number of real roots of $p_n$. In this paper, motivated by a question from Igor Pritsker, we prove that almost surely the following convergence holds: \begin{eqnarray*} \lim_{n\to\infty} \frac{N_n([-1,1])}{\log n} &=& \frac 1 π. \end{eqnarray*} This convergence could be viewed as a local strong law for the real roots. The main ingredient in the proof is a set of maximal inequalities that reduces the proof to proving convergence along lacunary subsequences, which in turn follows from a recent concentration estimate of Can--Nguyen.

math.PR↗

arXiv2021

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

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 non-trivial 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.

math.PR↗

arXiv2017

Positive sparse domination of variational Carleson operators

Due to its nonlocal nature, the $r$-variation norm Carleson operator $C_r$ does not yield to the sparse domination techniques of Lerner, Di Plinio and Lerner, Lacey. We overcome this difficulty and prove that the dual form to $C_r$ can be dominated by a positive sparse form involving $L^p$ averages. Our result strengthens the $L^p$ estimates by Oberlin et. al. As a corollary, we obtain quantitative weighted norm inequalities improving on previous results by Do and Lacey. Our proof relies on the localized outer $L^p$-embeddings of Di Plinio-Ou and Uraltsev.

math.CA↗