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Oanh Nguyen

Publications and source records attributed to Oanh Nguyen.

At least 19 recordsLinked to original sources
arXiv2026

Supercriticality of the SIRS on random networks

We study how long the SIRS process persists or how quickly it reaches extinction across various network topologies. Our results provide a three-part characterization of this process: In finite sparse graphs, we prove the existence of a regime where the process survives for an exponentially long time. In heavy-tailed networks with power-law-like exponents, we show that for all range of parameters, the survival time is exponential. Finally, for infinite trees, we find sufficient conditions for strong survival, showing the root is re-infected infinitely often even for light-tailed distributions like the Poisson distribution.

math.PR↗
arXiv2026

Convergence of higher derivatives of random polynomials with independent roots

Let $\mu$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $\mu$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for large classes of measures $\mu$, the empirical distribution of zeros of the $k$-th derivative converges back to $\mu$ for all derivative orders $k=o(n/\log n)$. This includes all discrete measures and a broad family of measures satisfying a mild dimension-nondegeneracy condition. We further establish a robustness result showing that, for arbitrary $\mu$, even after adding a vanishing proportion of roots drawn from a dimension-nondegenerate perturbation, the derivative zero measures still converge back to $\mu$. These results break the previously known logarithmic barrier on the order of differentiation and demonstrate that the limiting root distribution is preserved under differentiation of order growing nearly linearly with the degree.

math.PR↗
arXiv2025

On the First Non-Universal Term in Random Polynomial Real Zeros

Let $P_n(x) = \sum_{k=0}^{n} \xi_k x^k$ be a Kac random polynomial, where the coefficients $\xi_k$ are i.i.d.\ copies of a given random variable $\xi$. Based on numerical experiments, it has been conjectured that if $\xi$ has mean zero, unit variance, and a finite $(2+\varepsilon_0)$-moment for some $\varepsilon_0>0$, then \[ \mathbb{E}[N_{\mathbb{R}}(P_n)] \;=\; \frac{2}{\pi} \log n + C_{\xi} + o_n(1), \] where $N_{\mathbb{R}}(P_n)$ denotes the number of real roots of $P_n$, and $C_{\xi}$ is an absolute constant depending only on $\xi$, which is nonuniversal. Prior to this work, the existence of $C_{\xi}$ had only been established by Do-Nguyen-Vu (2015, \emph{Proc.\ Lond.\ Math.\ Soc.}) under the additional assumption that $\xi$ either admits a $(1+p)$-integrable density or is uniformly distributed on $\{\pm 1, \pm 2, \dots, \pm N\}$. In this paper, using a different method, we remove these extra conditions on $\xi$, and extend the result to the setting where the $\xi_k$ are independent but not necessarily identically distributed. Moreover, this proof strategy provides an alternative description of the constant $C_{\xi}$, and this new perspective serves as the key ingredient in establishing that $C_{\xi}$ depends continuously on the distribution of $\xi$.

math.PR↗
arXiv2024

Optimal bound for survival time of the SIRS process on star graphs

We analyze the Susceptible-Infected-Recovered-Susceptible (SIRS) process, a continuous-time Markov chain frequently employed in epidemiology to model the spread of infections on networks. In this framework, infections spread as infected vertices recover at rate 1, infect susceptible neighbors independently at rate $\lambda$, and recovered vertices become susceptible again at rate $\alpha$. This model presents a significantly greater analytical challenge compared to the SIS model, which has consequently inspired a much more extensive and rich body of mathematical literature for the latter. Understanding the survival time, the duration before the infection dies out completely, is a fundamental question in this context. On general graphs, survival time heavily depends on the infection's persistence around high-degree vertices (known as hubs or stars), as long persistence enables transmission between hubs and prolongs the process. In contrast, short persistence leads to rapid extinction, making the dynamics on star graphs, which serve as key representatives of hubs, particularly important to study. In the 2016 paper by Ferreira, Sander, and Pastor-Satorras, it was conjectured, based on intuitive arguments, that the survival time for SIRS on a star graph with $n$ leaves is bounded above by $(\lambda^2 n)^\alpha$ for large $n$. Later, in one of a few mathematically rigorous results for SIRS, Friedrich, G{\"o}bel, Klodt, Krejca, and Pappik provided an upper bound of $n^\alpha \log n$, with contains an additional $\log n$ and no dependence on $\lambda$. We resolve this conjecture by proving that the survival time is indeed of order $(\lambda^2 n)^\alpha$, with matching upper and lower bounds. Additionally, we show that this holds even in the case where only the root undergoes immunization, while the leaves revert to susceptibility immediately after recovery.

math.PR↗
arXiv2024

Subdifferential Calculus for Ordered Set-Valued Mappings between Infinite-Dimensional Spaces

The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various constraints in infinite dimensions. The main results establish new sum and chain rules for major subdifferential constructions associated with ordered set-valued mappings under appropriate qualification and sequentially normal compactness conditions.

math.OC↗
arXiv2024

Roots of random trigonometric polynomials with general dependent coefficients

We consider random trigonometric polynomials with general dependent coefficients. We show that under mild hypotheses on the structure of dependence, the asymptotics as the degree goes to infinity of the expected number of real zeros coincides with the independent case. To the best of our knowledge, this universality result is the first obtained in a non-Gaussian dependent context. Our proof highlights the robustness of real zeros, even in the presence of dependencies. These findings bring the behavior of random polynomials closer to real-world models, where dependencies between coefficients are common.

math.PR↗
arXiv2024

Concentration inequalities for the number of real zeros of Kac polynomials

We study concentration inequalities for the number of real roots of the classical Kac polynomials $$f_{n} (x) = \sum_{i=0}^n ξ_i x^i$$ where $ξ_i$ are independent random variables with mean 0, variance 1, and uniformly bounded $(2+\ep_0)$-moments. We establish polynomial tail bounds, which are optimal, for the bulk of roots. For the whole real line, we establish sub-optimal tail bounds.

math.PR↗
arXiv2023

Hole radii for the Kac polynomials and derivatives

The Kac polynomial $$f_n(x) = \sum_{i=0}^{n} ξ_i x^i$$ with independent coefficients of variance 1 is one of the most studied models of random polynomials. It is well-known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. In a beautiful work \cite{michelen2020real}, Michelen showed that the holes at $\pm 1$ are of order $1/n$. We show that in fact, all the hole radii are of the same order. The same phenomenon is established for the derivatives of the Kac polynomial as well.

math.PR↗
arXiv2023

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

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.

math.PR↗
arXiv2022

Real roots of random orthogonal polynomials with exponential weights

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 $W P_n.$

math.PR↗
arXiv2022

The number of limit cycles bifurcating from a randomly perturbed center

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 \emph{univariate} polynomial $\displaystyle f_n(x) = \sum_{m=0}^n c_m ξ_m x^m$ with independent coefficients $ξ_m$ having mean zero, variance 1 and $c_m \sim m^{-1/2}$. This polynomial belongs to the class of {\it generalized Kac polynomials} at the critical regime. We provide asymptotics for the average number of real zeros and answer the question on bifurcating limit cycles. Additionally, we provide the correct order of the mean number of real roots in the subcritical regime.

math.PR↗
arXiv2022

Subcritical epidemics on random graphs

We study the contact process on random graphs with low infection rate $\lambda$. For random $d$-regular graphs, it is known that the survival time is $O(\log n)$ below the critical $\lambda_c$. By contrast, on the Erd\H{o}s-R\'enyi random graphs $\mathcal G(n,d/n)$, rare high-degree vertices result in much longer survival times. We show that the survival time is governed by high-density local configurations. In particular, we show that there is a long string of high-degree vertices on which the infection lasts for time $n^{\lambda^{2+o(1)}}$. To establish a matching upper bound, we introduce a modified version of the contact process which ignores infections that do not lead to further infections and allows for a shaper recursive analysis on branching process trees, the local-weak limit of the graph. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments.

math.PR↗
arXiv2021

Roots of random functions: A framework for local universality

We investigate the local distribution of roots of random functions of the form $F_n(z)= \sum_{i=1}^n ξ_i ϕ_i(z) $, where $ξ_i$ are independent random variables and $ϕ_i (z) $ are arbitrary analytic functions. Starting with the fundamental works of Kac and Littlewood-Offord in the 1940s, random functions of this type have been studied extensively in many fields of mathematics. We develop a robust framework to solve the problem by reducing, via universality theorems, the calculation of the distribution of the roots and the interaction between them to the case where $ξ_i$ are gaussian. In this special case, one can use the Kac-Rice formula and various other tools to obtain precise answers. Our framework has a wide range of applications, which include the most popular models of random functions, such as random trigonometric polynomials and all basic classes of random algebraic polynomials (Kac, Weyl, and elliptic). Each of these ensembles has been studied heavily by deep and diverse methods. Our method, for the first time, provides a unified treatment of all of them. Among the applications, we derive the first local universality result for random trigonometric polynomials with arbitrary coefficients. When restricted to the study of real roots, this result extends several recent results, proved for less general ensembles. For random algebraic polynomials, we strengthen several recent results of Tao and the second author, with significantly simpler proofs. As a corollary, we sharpen a classical result of Erd{ö}s and Offord on real roots of Kac polynomials, providing an optimal error estimate. Another application is a refinement of a recent result of Flasche and Kabluchko on the roots of random Taylor series.

math.PR↗
arXiv2020

Random polynomials: central limit theorems for the real roots

The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood-Offord and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova's theorem.

math.PR↗
arXiv2020

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

In this paper, we study the number of real roots of random trigonometric polynomials with iid coefficients. When the coefficients have zero mean, unit variance and some finite high moments, we show that the variance of the number of real roots is asymptotically linear in terms of the expectation; furthermore, the multiplicative constant in this linear relationship depends only on the kurtosis of the common distribution of the polynomial's coefficients. This result is in sharp contrast to the classical Kac polynomials whose corresponding variance depends only on the first two moments. Our result is perhaps the first paper to establish the variance for general distribution of the coefficients including discrete ones, for a model of random polynomials outside the family of the Kac polynomials. Our method gives a fine comparison framework throughout Edgeworth expansion, asymptotic Kac-Rice formula and a detailed analysis of characteristic functions.

math.PR↗
arXiv2020

Survival and extinction of epidemics on random graphs with general degrees

In this paper, we establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $λ_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution $ξ$ has an exponential tail, i.e., $\mathbb{E} e^{cξ}<\infty$ for some $c>0$, settling a conjecture by Huang and Durrett [12]. On the random graph with degree distribution $μ$, we show that if $μ$ has an exponential tail, then for small enough $λ$ the contact process with the all-infected initial condition survives for $n^{1+o(1)}$-time w.h.p. (short survival), while for large enough $λ$ it runs over $e^{Θ(n)}$-time w.h.p. (long survival). When $μ$ is subexponential, we prove that the contact process w.h.p. displays long survival for any fixed $λ>0$.

math.PR↗
arXiv2019

On the spectrum of random walks on complete finite $d$-ary trees

In the present paper, we determine the full spectrum of the simple random walk on finite, complete $d$-ary trees. We also find an eigenbasis for the transition matrix. As an application, we apply our results to get a lower bound for the interchange process on complete, finite d-ary trees, which we conjecture to be sharp.

math.PR↗