About

Yen Do is an Associate Professor in the Department of Mathematics at the University of Virginia. His research focuses on Harmonic Analysis, Random Polynomials, Analysis, Probability, and related areas. He is involved in teaching and mentoring students, and maintains a personal webpage with additional information about his work and interests.

Research topics

• Mathematics
• Pure mathematics
• Combinatorics
• Mathematical analysis
• Discrete mathematics

Selected publications

• Central Limit Theorem for the number of real roots of random orthogonal polynomials

  Annales de l Institut Henri Poincaré Probabilités et Statistiques · 2024-07-31

  preprint1st authorCorresponding

  In this note we study the number of real roots of a wide class of random orthogonal polynomials with gaussian coefficients. Using the method of Wiener Chaos we show that the fluctuation in the bulk is asymptotically gaussian, even when the local correlations are not necessarily the same.

  PublisherDOI

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

  arXiv (Cornell University) · 2024-03-11

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Real roots of random polynomials: asymptotics of the variance

  arXiv (Cornell University) · 2023-03-09

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Random orthonormal polynomials: Local universality and expected number of real roots

  Transactions of the American Mathematical Society · 2023-01-25 · 6 citations

  article1st authorCorresponding

  We consider random orthonormal polynomials <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript n Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript i equals 0 Overscript n Endscripts xi Subscript i Baseline p Subscript i Baseline left-parenthesis x right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:munderover> <mml:msub> <mml:mi> ξ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} F_{n}(x)=\sum _{i=0}^{n}\xi _{i}p_{i}(x), \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi 0"> <mml:semantics> <mml:msub> <mml:mi> ξ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\xi _{0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , …, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi Subscript n"> <mml:semantics> <mml:msub> <mml:mi> ξ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\xi _{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are independent random variables with zero mean, unit variance and uniformly bounded <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 plus epsilon right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi> ε </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(2+\varepsilon )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> moments, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p Subscript n Baseline right-parenthesis Subscript n equals 0 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">(p_n)_{n=0}^{\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the system of orthonormal polynomials with respect to a fixed compactly supported measure on the real line. Under mild technical assumptions satisfied by many classes of classical polynomial systems, we establish universality for the leading asymptotics of the average number of real roots of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript n"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , both globally and locally. Prior to this paper, these results were known only for random orthonormal polynomials with Gaussian coefficients (see D. D. Lubinsky, I. E. Pritsker, and X. Xie [Proc. Amer. Math. Soc. 144 (2016), pp. 1631–1642]) using the Kac-Rice formula, a method that does not extend to the generality of our paper.

  PublisherDOI

• Random trigonometric polynomials: Universality and non-universality of the variance for the number of real roots

  Annales de l Institut Henri Poincaré Probabilités et Statistiques · 2022-07-15 · 12 citations

  article1st authorCorresponding

  Dans cet article, nous étudions le nombre de zéros réels de polynômes trigonométriques avec coefficients i.i.d. Quand les coefficients sont centrés, réduits, et possèdent des moments finis d’ordre suffisamment élevé, nous montrons que la variance du nombre de zéros est asymptotiquement linéaire en son espérance ; de plus, la constante multiplicative dans cette relation linéaire dépend seulement du kurtosis de la loi commune des coefficients du polynôme. Ce résultat contraste fortement avec les classiques polynômes de Kac pour lesquels la variance ne dépend que des deux premiers moments. Il s’agit probablement du premier résultat sur ce type de questions pour des lois générales des coefficients, y compris des lois discrètes, pour des polynômes qui ne sont pas dans la famille des polynômes de Kac. L’expansion de Edgeworth, la formule asymptotique de Kac–Rice et l’analyse précise des fonctions caractéristiques sont les outils principaux de notre approche.

  PublisherDOI

• Real roots of random orthogonal polynomials with exponential weights

  SHAREOK (University of Oklahoma) · 2022-12-30

  preprintOpen access1st authorCorresponding

  We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}ξ_{i}p_{i}(x), $$ where $ξ_{0}$, . . . , $ξ_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and $\{p_n\}_{n=0}^{\infty}$ is the system of orthonormal polynomials with respect to a general exponential weight $W$ on the real line. This class of orthogonal polynomials includes the popular Hermite and Freud polynomials. We establish universality for the leading asymptotics of the expected number of real roots of $P_n$, both globally and locally. In addition, we find an almost sure limit of the measures counting all roots of $P_n.$ This is accomplished by introducing new ideas on applications of the inverse Littlewood-Offord theory in the context of the classical three term recurrence relation for orthogonal polynomials to establish anti-concentration properties, and by adapting the universality methods to the weighted random orthogonal polynomials of the form $W P_n.$

  PublisherDOI

• Generalized Carleson embeddings into weighted outer measure spaces

  Journal of Mathematical Analysis and Applications · 2021-09-24

  article1st authorCorresponding
  PublisherDOI

• Central Limit Theorem for the number of real roots of random orthogonal polynomials

  arXiv (Cornell University) · 2021-11-17 · 1 citations

  preprintOpen access1st authorCorresponding

  In this note we study the number of real roots of a wide class of random orthogonal polynomials with gaussian coefficients. Using the method of Wiener Chaos we show that the fluctuation in the bulk is asymptotically gaussian, even when the local correlations are not necessarily the same.

  PublisherOA PDFDOI

• Random orthonormal polynomials: local universality and expected number of real roots

  arXiv (Cornell University) · 2020-12-20

  preprintOpen access1st authorCorresponding

  We consider random orthonormal polynomials $$ F_{n}(x)=\sum_{i=0}^{n}ξ_{i}p_{i}(x), $$ where $ξ_{0}$, \dots, $ξ_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep)$ moments, and $(p_n)_{n=0}^{\infty}$ is the system of orthonormal polynomials with respect to a fixed compactly supported measure on the real line. Under mild technical assumptions satisfied by many classes of classical polynomial systems, we establish universality for the leading asymptotics of the average number of real roots of $F_n$, both globally and locally. Prior to this paper, these results were known only for random orthonormal polynomials with Gaussian coefficients \cite{lubinsky2016linear} using the Kac-Rice formula, a method that does not extend to the generality of our paper.

  PublisherOA PDFDOI

• Central Limit Theorems for the Real Zeros of Weyl Polynomials

  American Journal of Mathematics · 2020-01-01 · 13 citations

  article1st authorCorresponding

  We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=\xi_0+\xi_1 x+\cdots+{1\over\sqrt{n!}}\xi_n x^n$, where $\xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws and repulsion properties of these real roots.

  PublisherDOI

Recent grants

• Fourier analysis and applications to completely integrable systems

  NSF · $134k · 2012–2015

• Fourier analysis and applications to completely integrable systems

  NSF · $30k · 2014–2016

• Topics in Harmonic Analysis and Probabilistic Analysis

  NSF · $154k · 2018–2022

Frequent coauthors

• Hoi H. Nguyen

  Bạch Mai Hospital

  18 shared

• Van Vu

  13 shared

• Christoph Thiele

  6 shared

• Richard Oberlin

  Florida State University

  5 shared

• Igor E. Pritsker

  Oklahoma State University

  4 shared

• Eyvindur A. Palsson

  4 shared

• Michael T. Lacey

  4 shared

• Camil Muscalu

  Romanian Academy

  4 shared

Education

• PhD, Mathematics

  University of California at Los Angeles

  2010