arXiv subjects
Publications and source records attributed to Hoi H. Nguyen.
We show that the number of real roots of random trigonometric polynomials with i.i.d. coefficients, which are either bounded or satisfy the logarithmic Sobolev inequality, satisfies an exponential concentration of measure.
In this note we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank of random matrices over F_p and the equi-distribution behavior of normal vectors of random hyperplanes. We also study the probability that a random square matrix is eigenvalue-free, or when its characteristic polynomial is divisible by a given irreducible polynomial in the limit n to infinity in F_p. We show that these statistics are universal, extending results of Stong and Neumann-Praeger beyond the uniform model.
We address overcrowding estimates for the singular values of random iid matrices, as well as for the eigenvalues of random Wigner matrices. We show evidence of long range separation under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds
We study the distribution of the cokernels of adjacency matrices (the Smith groups) of certain models of random $r$-regular graphs and directed graphs, using recent mixing results of Mészáros. We explain how convergence of such distributions to a limiting probability distribution implies asymptotic nonsingularity of the matrices, giving another perspective on recent results of Huang and Mészáros on asymptotic nonsingularity of adjacency matrices of random regular directed and undirected graphs, respectively. We also remark on the new distributions on finite abelian groups that arise, in particular in the $p$-group aspect when $p\mid r$.
For a random matrix of entries sampled independently from a fairly general distribution in Z we study the probability that the cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that the linear map between the integer lattices given by the matrix is surjective. We show that these statistics are asymptotically universal (as the size of the matrix goes to infinity), given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra, even when the distribution of matrix entries is very distorted. Our method is robust and works for Laplacians of random digraphs and sparse matrices with the probability of an entry non-zero only n^{-1+epsilon}.
We show that a nearly square iid random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz. Our result extends to sparse matrices as well as to matrices of dependent entries.
It is well-known that distances in random iid matrices are highly concentrated around their mean. In this note we extend this concentration phenomenon to Wigner matrices. Exponential bounds for the lower tail are also included.
We provide a characterization for anti-concentration of inhomogeneous random walks in non-abelian groups. In application we extend the classical bounds by Erdos-Littlewood-Offord and Sarkozy-Szemeredi to non-abelian settings.
Suppose that A_1,\dots, A_N are independent random matrices whose atoms are iid copies of a random variable ξof mean zero and variance one. It is known from the works of Newman et. al. in the late 80s that when ξis gaussian then N^{-1} \log ||A_N \dots A_1|| converges to a non-random limit. We extend this result to more general matrices with explicit rate of convergence. Our method relies on a simple connection between structures and dynamics.
In this short note we address a gaussian property of normal vectors in random non-Hermitian matrices. The approach uses a simple geometric and comparison technique.
Let v_1,...,v_{n-1} be n-1 independent vectors in R^n (or C^n). We study x, the unit normal vector of the hyperplane spanned by the v_i. Our main finding is that x resembles a random vector chosen uniformly from the unit sphere, under some randomness assumption on the v_i. Our result has applications in random matrix theory. Consider an n by n random matrix with iid entries. We first prove an exponential bound on the upper tail for the least singular value, improving the earlier linear bound by Rudelson and Vershynin. Next, we derive optimal delocalization for the eigenvectors corresponding to eigenvalues of small modulus.
Let $X$ be a matrix sampled uniformly from the set of doubly stochastic matrices of size $n\times n$. We show that the empirical spectral distribution of the normalized matrix $\sqrt{n}(X-{\mathbf {E}}X)$ converges almost surely to the circular law. This confirms a conjecture of Chatterjee, Diaconis and Sly.
Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_n|$ satisfies a central limit theorem. More precisely, \begin{eqnarray*}\sup_{x\in {\mathbf {R}}}\biggl|{\mathbf {P}}\biggl(\frac{\log(|\det A_n|)-({1}/{2})\log (n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf {P}}\bigl(\mathbf {N}(0,1)\le x\bigr)\biggr|\\\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}
Let $ξ$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 ξ_1 + ... + a_n ξ_n $$ where $ξ_i$ are iid copies of $ξ$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.
Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that $M_n$ is non-singular with probability $1-O(n^{-C})$ for any positive constant $C$. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.
Let $M_n$ be a random matrix of size $n\times n$ and let $λ_1,...,λ_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $μ_{M_n}$ of $M_n$ is defined as $$μ_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(λ_k)\le s; \Im(λ_k)\le t\}.$$ The circular law theorem in random matrix theory asserts that if the entries of $M_n$ are i.i.d. copies of a random variable with mean zero and variance $σ^2$, then the empirical spectral distribution of the normalized matrix $\frac{1}{σ\sqrt{n}}M_n$ of $M_n$ converges almost surely to the uniform distribution $μ_\cir$ over the unit disk as $n$ tends to infinity. In this paper we show that the empirical spectral distribution of the normalized matrix of $M_n$, a random matrix whose rows are independent random $(-1,1)$ vectors of given row-sum $s$ with some fixed integer $s$ satisfying $|s|\le (1-o(1))n$, also obeys the circular law. The key ingredient is a new polynomial estimate on the least singular value of $M_n$.
It is shown that a random $(0,1)$ matrix whose rows are independent random vectors of exactly $n/2$ zero components is non-singular with probability $1-O(n^{-C})$ for any $C>0$. The proof uses a non-standard inverse-type Littlewood-Offord result.
A sequence $A$ of elements an additive group $G$ is {\it incomplete} if there exists a group element that {\it can not} be expressed as a sum of elements from $A$. The study of incomplete sequences is a popular topic in combinatorial number theory. However, the structure of incomplete sequences is still far from being understood, even in basic groups. The main goal of this paper is to give a characterization of incomplete sequences in the vector space $F_p^d$, where $d$ is a fixed integer and $p$ is a large prime. As an application, we give a new proof for a recent result by Gao-Ruzsa-Thangadurai on the Olson's constant of $\F_p^2$ and partially answer their conjecture concerning $F_p^3$.