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Hoi H. Nguyen

Publications and source records attributed to Hoi H. Nguyen.

At least 19 recordsLinked to original sources
arXiv2026

Central limit theorem for real zeros of random Weyl polynomials with general coefficients

For a random polynomial, the number of real zeros $N_{\mathbb R}$ is a highly nonlinear function of its coefficients, and its statistical properties have been studied extensively. A natural question is whether $N_{\mathbb R}$ satisfies a central limit theorem. For various ensembles with iid standard Gaussian coefficients, such central limit theorems have been established; see, for instance, arXiv:1911.12182, arXiv:1801.06331, arXiv:1401.5745; Azais and Leon, Electron. J. Probab. 18 (2013), no. 68; arXiv:1504.05355, arXiv:2111.09015, arXiv:1707.09276, arXiv:1005.4113. These results rely on a rich range of tools, including Kac-Rice formulas, moment methods, and Wiener chaos decompositions. In the non-Gaussian setting, however, many of these tools are unavailable. To the best of our knowledge, prior central limit theorems beyond the Gaussian setting were limited to Kac-type polynomials, including hyperbolic polynomials; see the works of Maslova (1974), O. Nguyen and Vu (arXiv:1904.04347), and, more recently, Do, N. Nguyen, and O'Rourke (arXiv:2605.26402). In this paper, we prove a central limit theorem for the total number of real zeros of Weyl polynomials whose coefficients are iid copies of a symmetric, mean-zero, variance-one subgaussian random variable $ξ$. This substantially extends one of the main results of Do and Vu (arXiv:1707.09276) to a broad class of non-Gaussian distributions, including the Rademacher distribution. Without the symmetry assumption, we prove central limit theorems for the number of real zeros for positive bulk intervals, as well as for $[0,\infty)$. Our proof combines the uniform one-point anti-concentration estimates from our recent work (arXiv:2511.07735) with the localization of Weyl polynomials around the coefficient index $i\approx x^2$. While our proofs use comparison to compute the variances, the CLT deduction is rather direct.

math.PR↗
arXiv2026

Rank fluctuations of matrix products and a moment method for growing groups

We consider the cokernel $G_n = \mathbf{Cok}(A_{k} \cdots A_2 A_1)$ of a product of independent $n \times n$ random integer matrices with iid entries from generic nondegenerate distributions, in the regime where both $n$ and $k$ are sent to $\infty$ simultaneously. In this regime we show that the cokernel statistics converge universally to the reflecting Poisson sea, an interacting particle system constructed in arXiv:2312.11702, at the level of $1$-point marginals. In particular, $\operatorname{corank}(A_{k} \cdots A_2 A_1 \pmod{p}) \sim \log_p k$, and its fluctuations are $O(1)$ and converge to a discrete random variable defined in arXiv:2310.12275. The main difference with previous works studying cokernels of random matrices is that $G_n$ does not converge to a random finite group; for instance, the $p$-rank of $G_n$ diverges. This means that the usual moment method for random groups does not apply. Instead, we proceed by proving a `rescaled moment method' theorem applicable to a general sequence of random groups of growing size. This result establishes that fluctuations of $p$-ranks and other statistics still converge to limit random variables, provided that certain rescaled moments $\mathbb{E}[\#\operatorname{Hom}(G_n,H)]/C(n,H)$ converge.

math.PR↗
arXiv2026

The anti-concentration phenomenon with respect to random permutations

The anti-concentration phenomenon in probability theory has been intensively studied in recent years, with applications across many areas of mathematics. In most existing works, the ambient probability space is a product space generated by independent random variables. In this paper, we initiate a systematic study of anti-concentration when the ambient space is the symmetric group, equipped with the uniform measure. Concretely, we focus on the random sum $S_π = \sum_{i=1}^{n} w_i\, v_{π(i)}$, where $w=(w_1,\dots,w_n)$ and $v=(v_1,\dots,v_n)$ are fixed vectors and $π$ is a uniformly random permutation. The paper contains several new results, addressing both discrete and continuous anti-concentration phenomena. On the discrete side, we establish a near-optimal structural characterization of the vectors $w$ and $v$ under the assumption that the concentration probability $\sup_x P(S_π=x)$ is polynomially large. On the continuous side, we study the small-ball event $|S_π-L|\le δ$. Our results exhibit sub-gaussian decay in $L$. Our results have applications in various areas. First, we use our inverse theorems to derive and strengthen a number of previous anti-concentration bounds. In particular, we show that if both $w$ and $v$ have distinct entries, then $\sup_x P(S_π=x) \le n^{-5/2+o(1)}$. Next, we apply our new results to study random polynomials, and prove that the number of extremal points of random permutation polynomials is bounded by $O(\log n)$, extending results of S{ö}ze~\cite{Soze1, Soze2}. In the final application, we prove that random matrices whose rows are independent random permutations of a fixed non-degenerate vector are nonsingular with high probability.

math.CO↗
arXiv2025

Real Roots of Random Weyl Polynomials with General Coefficients: Expectation and Variance

In this paper, we investigate the number of real zeros of random Weyl polynomials of degree \(n \to \infty\) with general coefficient distributions. Motivated by the results of arXiv:1409.4128 and arXiv:1402.4628 as well as arXiv:1711.03316 and arXiv:1912.11901, we determine how the expected number of real zeros and their variance, over various natural intervals, depend on the moments of the common coefficient distribution. Our main finding is that while the first-order asymptotic of the expectation is universal, the next-order correction depends on the third and fourth moments of the distribution, and may grow linearly with \(\log n\), depending on the interval under consideration. In contrast, for the variance we show that the leading-order term is universal, which differs from the behavior observed for random trigonometric polynomials in arXiv:1711.03316 and arXiv:1912.11901. Our approach relies on an Edgeworth expansion for random walks arising from Weyl polynomials, a result of independent interest.

math.PR↗
arXiv2025

Cohen-Lenstra flag universality for random matrix products

For $n \times n$ random integer matrices $M_1,\ldots,M_k$, the cokernels of the partial products $\mathrm{cok}(M_1 \cdots M_i), 1 \leq i \leq k$ naturally define a random flag of abelian $p$-groups. We prove that as $n \to \infty$, this flag converges universally, for any nondegenerate entry distribution, to the Cohen-Lenstra type measure which weights each flag inversely proportional to the size of its automorphism group. As a corollary, we prove universality of certain formulas for the limiting conditional distribution of $\mathrm{cok}(M_1M_2)$ given $\mathrm{cok}(M_1),\mathrm{cok}(M_2)$ in terms of Hall-Littlewood structure constants, which were previously obtained only for Haar matrices over $\mathbb{Z}_p$. Our proofs combine the general technology of Sawin-Wood, matrix product moment computations following those of Nguyen-Van Peski, and the computation done previously for Haar $p$-adic matrices by Huang.

math.PR↗
arXiv2025

Eigenvalue gaps of the Laplacian of random graphs

We show that, with very high probability, the random graph Laplacian has simple spectrum. Our method provides a quantitatively effective estimate of the spectral gaps. Along the way, we establish results on affine no-gaps delocalization, no-structure delocalization, overcrowding and small entries of the eigenvectors for the Laplacian model. These findings are of independent interest.

math.PR↗
arXiv2024

Concentration of the number of real roots of random polynomials

Many statistics of roots of random polynomials have been studied in the literature, but not much is known on the concentration aspect. In this note we present a systematic study of this question, aiming towards nearly optimal bounds to some extent. Our method is elementary and works well for many models of random polynomials, with gaussian or non-gaussian coefficients.

math.PR↗
arXiv2023

Universality for cokernels of random matrix products

For random integer matrices $M_1,\ldots,M_k \in \operatorname{Mat}_n(\mathbb{Z})$ with independent entries, we study the distribution of the cokernel $\operatorname{cok}(M_1 \cdots M_k)$ of their product. We show that this distribution converges to a universal one as $n \to \infty$ for a general class of matrix entry distributions, and more generally show universal limits for the joint distribution of $\operatorname{cok}(M_1),\operatorname{cok}(M_1M_2),\ldots,\operatorname{cok}(M_1 \cdots M_k)$. Furthermore, we characterize the universal distributions arising as marginals of a natural generalization of the Cohen-Lenstra measure to sequences of abelian groups with maps between them, which weights sequences inversely proportionally to their number of automorphisms. The proofs develop an extension of the moment method of Wood to joint moments of multiple groups, and rely also on the connection to Hall-Littlewood polynomials and symmetric function identities. As a corollary we obtain an explicit universal distribution for coranks of random matrix products over $\mathbb{F}_p$ as the matrix size tends to infinity.

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

A note on the singularity probability of random directed $d$-regular graphs

In this note we show that the singular probability of the adjacency matrix of a random $d$-regular graph on $n$ vertices, where $d$ is fixed and $n \to \infty$, is bounded by $n^{-1/3+o(1)}$. This improves a recent bound by Huang. Our method is based on the study of the singularity problem modulo a prime together with an inverse-type result on the decay of the characteristic function. The latter is related to the inverse Kneser's problem in combinatorics.

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

Local and global universality of random matrix cokernels

In this paper we study the cokernels of various random integral matrix models, including random symmetric, random skew-symmetric, and random Laplacian matrices. We provide a systematic method to establish universality under very general randomness assumption. Our highlights include both local and global universality of the cokernel statistics of all these models. In particular, we find the probability that a sandpile group of an Erdos-Renyi random graph is cyclic, answering a question of Lorenzini from 2008.

math.PR↗
arXiv2022

Universality of Poisson limits for moduli of roots of Kac polynomials

We give a new proof of a recent resolution by Michelen and Sahasrabudhe of a conjecture of Shepp and Vanderbei that the moduli of roots of Gaussian Kac polynomials of degree $n$, centered at $1$ and rescaled by $n^2$, should form a Poisson point process. We use this new approach to verify a conjecture of Michelen and Sahasrabudhe that the Poisson statistics are in fact universal.

math.PR↗
arXiv2021

Universality of the minimum modulus for random trigonometric polynomials

It has been shown in a recent work by Yakir-Zeitouni that the minimum modulus of random trigonometric polynomials with Gaussian coefficients has a limiting exponential distribution. We show this is a universal phenomenon. Our approach relates the joint distribution of small values of the polynomial at a fixed number $m$ of points on the circle to the distribution of a certain random walk in a $4m$-dimensional phase space. Under Diophantine approximation conditions on the angles, we obtain strong small ball estimates and a local central limit theorem for the distribution of the walk.

math.PR↗
arXiv2021

A note on inverse results of random walks in Abelian groups

In this short note we give various near optimal characterizations of random walks over finite Abelian groups with large maximum discrepancy from the uniform measure. We also provide several interesting connections to existing results in the literature.

math.CO↗
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↗