### Results for "Van Vu"

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Random matrices have simple spectrumDec 03 2014Let $M_n = (\xi_{ij})_{1 \leq i,j \leq n}$ be a real symmetric random matrix in which the upper-triangular entries $\xi_{ij}, i<j$ and diagonal entries $\xi_{ii}$ are independent. We show that with probability tending to 1, $M_n$ has no repeated eigenvalues. ... More
The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matricesJan 29 2011Sep 17 2011A well known conjecture of Wigner, Dyson, and Mehta asserts that the (appropriately normalized) $k$-point correlation functions of the eigenvalues of random $n \times n$ Wigner matrices in the bulk of the spectrum converge (in various senses) to the $k$-point ... More
Random covariance matrices: Universality of local statistics of eigenvaluesDec 07 2009May 25 2012We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of a large rectangular matrix $M=M_{n,p}=(\zeta_{ij})_{1\leq i\leq p;1\leq j\leq n}$ whose entries are i.i.d. random variables of mean zero, variance one, and having finite $C_0$th moment ... More
Random perturbation of low rank matrices: Improving classical boundsNov 12 2013Nov 16 2017Matrix perturbation inequalities, such as Weyl's theorem (concerning the singular values) and the Davis-Kahan theorem (concerning the singular vectors), play essential roles in quantitative science; in particular, these bounds have found application in ... More
Non-abelian Littlewood-Offord inequalitiesJun 05 2015Aug 06 2015In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. ... More
Singular vectors under random perturbationApr 12 2010Computing the first few singular vectors of a large matrix is a problem that frequently comes up in statistics and numerical analysis. Given the presence of noise, exact calculation is hard to achieve, and the following problem is of importance: \vskip2mm ... More
Sum-product estimates via directed expandersMay 04 2007Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$ for which the ... More
A structural approach to subset-sum problemsApr 20 2008We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the paper. These results have applications in various areas, such ... More
Central limit theorems for random polytopes in a smooth convex setMar 24 2005Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n$ random points in $K$ independently according to the uniform distribution. The convex hull of these points, denoted by $K_n$, is called a {\it random polytope}. We prove that several ... More
A simple SVD algorithm for finding hidden partitionsApr 15 2014Finding a hidden partition in a random environment is a general and important problem, which contains as subproblems many famous questions, such as finding a hidden clique, finding a hidden coloring, finding a hidden bipartition etc. In this paper, we ... More
Dictionary Learning with Few Samples and Matrix ConcentrationMar 30 2015Let $A$ be an $n \times n$ matrix, $X$ be an $n \times p$ matrix and $Y = AX$. A challenging and important problem in data analysis, motivated by dictionary learning and other practical problems, is to recover both $A$ and $X$, given $Y$. Under normal ... More
Local resilience of graphsJun 27 2007Dec 01 2007In this paper, we initiate a systematic study of graph resilience. The (local) resilience of a graph G with respect to a property P measures how much one has to change G (locally) in order to destroy P. Estimating the resilience leads to many new and ... More
Random matrices: Universality of local spectral statistics of non-Hermitian matricesJun 09 2012Mar 16 2015It is a classical result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on ... More
Random matrices: The Universality phenomenon for Wigner ensemblesFeb 01 2012In this paper, we survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the ... More
A central limit theorem for the determinant of a Wigner matrixNov 27 2011Mar 29 2012We establish a central limit theorem for the log-determinant $\log|\det(M_n)|$ of a Wigner matrix $M_n$, under the assumption of four matching moments with either the GUE or GOE ensemble. More specifically, we show that this log-determinant is asymptotically ... More
Random matrices: Localization of the eigenvalues and the necessity of four momentsMay 17 2010Aug 12 2011Consider the eigenvalues $\lambda_i(M_n)$ (in increasing order) of a random Hermitian matrix $M_n$ whose upper-triangular entries are independent with mean zero and variance one, and are exponentially decaying. By Wigner's semicircular law, one expects ... More
A sharp inverse Littlewood-Offord theoremFeb 13 2009Oct 20 2009Let $\eta_i, i=1,..., n$ be iid Bernoulli random variables. Given a multiset $\bv$ of $n$ numbers $v_1, ..., v_n$, the \emph{concentration probability} $\P_1(\bv)$ of $\bv$ is defined as $\P_1(\bv) := \sup_{x} \P(v_1 \eta_1+ ... v_n \eta_n=x)$. A classical ... More
Random matrices: Universal properties of eigenvectorsMar 14 2011May 09 2011The four moment theorem asserts, roughly speaking, that the joint distribution of a small number of eigenvalues of a Wigner random matrix (when measured at the scale of the mean eigenvalue spacing) depends only on the first four moments of the entries ... More
Spectrum of complex networksSep 14 2018Sep 24 2018The study of complex networks has been one of the most active fields in science in recent decades. Spectral properties of networks (or graphs that represent them) are of fundamental importance. Researchers have been investigating these properties for ... More
Roots of random functions: A general condition for local universalityNov 09 2017Jan 23 2018We investigate the local distribution of roots of random functions of the form $F_n(z)= \sum_{i=1}^n \xi_i \phi_i(z)$, where $\xi_i$ are independent random variables and $\phi_i (z)$ are arbitrary analytic functions. Starting with the fundamental works ... More
Offloading Energy Efficiency with Delay Constraint for Cooperative Mobile Edge Computing NetworksNov 30 2018We propose a novel edge computing network architecture that enables edge nodes to cooperate in sharing computing and radio resources to minimize the total energy consumption of mobile users while meeting their delay requirements. To find the optimal task ... More
Smooth analysis of the condition number and the least singular valueMay 20 2008Aug 10 2009Let $\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate ... More
A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner MatricesJan 13 2011The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct aymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, ... More
Random matrices: The Four Moment Theorem for Wigner ensemblesDec 08 2011We survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner random matrix ensembles, focusing in particular on the Four Moment Theorem and its applications.
Random matrices: Universality of local eigenvalue statisticsJun 02 2009Jun 29 2010In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the ... More
Random matrices: Universality of local eigenvalue statistics up to the edgeAug 13 2009Jan 11 2010This is a continuation of our earlier paper on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in that paper from the bulk of the spectrum up to the edge. In particular, ... More
Random matrices: The distribution of the smallest singular valuesMar 03 2009Let $\a$ be a real-valued random variable of mean zero and variance 1. Let $M_n(\a)$ denote the $n \times n$ random matrix whose entries are iid copies of $\a$ and $\sigma_n(M_n(\a))$ denote the least singular value of $M_n(\a)$. ($\sigma_n(M_n(\a))^2$ ... More
On the singularity probability of random Bernoulli matricesJan 20 2005Aug 06 2008Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability 1/2). We show that the probability that $M_n$ is singular is at most $(3/4 +o(1))^n$, improving ... More
Random weighted projections, random quadratic forms and random eigenvectorsJun 13 2013Aug 16 2014We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms; (2) The infinity norm of most unit eigenvectors of a random $\pm ... More Classification theorems for sumsets modulo a primeNov 09 2008Jan 27 2009Let$\Z/pZ$be the finite field of prime order$p$and$A$be a subsequence of$\Z/pZ$. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of$A$? (2) When can one represent ... More Random Matrices: The circular LawAug 21 2007Feb 29 2008Let$\a$be a complex random variable with mean zero and bounded variance$\sigma^{2}$. Let$N_{n}$be a random matrix of order$n$with entries being i.i.d. copies of$\a$. Let$\lambda_{1}, ..., \lambda_{n}$be the eigenvalues of$\frac{1}{\sigma \sqrt ... More
Squares in sumsetsNov 09 2008Oct 29 2009A finite set $A$ of integers is square-sum-free if there is no subset of $A$ sums up to a square. In 1986, Erd\H os posed the problem of determining the largest cardinality of a square-sum-free subset of $\{1, ..., n \}$. Answering this question, we show ... More
From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matricesOct 16 2008Jan 01 2009The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\frac{1}{\sqrt{n}} M_n$ converges almost surely ... More
Sparse Random Matrices have Simple SpectrumFeb 10 2018Feb 18 2018Let $M_n$ be a class of symmetric sparse random matrices, with independent entries $M_{ij} = \delta_{ij} \xi_{ij}$ for $i \leq j$. $\delta_{ij}$ are i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq n^{-1+\delta}$ for any ... More
Central limit theorems for the real zeros of Weyl polynomialsJul 28 2017Aug 01 2017We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=xi_0 + xi_1 x+ ... + xi_n (n!)^{(-1/2)} x^n$, where $xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the ... More
Random matrices: Probability of NormalityNov 08 2017Feb 05 2019In this paper, we investigate the following question: How often is a random matrix normal? We consider a random $n\times n$ matrix, $M_n$, whose entries are i.i.d. Rademacher random variables (taking values $\{ \pm1 \}$ with probability $1/2$) and prove ... More
Random walks with different directions: Drunkards beware !Sep 29 2014As an extension of Polya's classical result on random walks on the square grids ($\Z^d$), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability ... More
The condition number of a randomly perturbed matrixMar 11 2007Let $M$ be an arbitrary $n$ by $n$ matrix. We study the condition number a random perturbation $M+N_n$ of $M$, where $N_n$ is a random matrix. It is shown that, under very general conditions on $M$ and $M_n$, the condition number of $M+N_n$ is polynomial ... More
On random $\pm 1$ matrices: Singularity and DeterminantNov 04 2004Jun 30 2008This papers contains two results concerning random $n \times n$ Bernoulli matrices. First, we show that with probability tending to one the determinant has absolute value $\sqrt {n!} \exp(O(\sqrt(n log n)))$. Next, we prove a new upper bound $.939^n$ ... More
Smooth analysis of the condition number and the least singular valueMay 20 2008May 19 2017Let $\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate ... More
Packing perfect matchings in random hypergraphsJun 30 2016Jul 05 2016We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as \emph{online sprinkling}. As an illustrative application of this method, we show that for any fixed integer $k\geq 3$, the binomial $k$-uniform random ... More
Random polynomials: central limit theorems for the real rootsApr 08 2019The 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. In this paper, we establish the Central Limit Theorem for the number of ... More
Random matrices: Sharp concentration of eigenvaluesJan 23 2012Aug 10 2013Let $W_n= \frac{1}{\sqrt n} M_n$ be a Wigner matrix whose entries have vanishing third moment, normalized so that the spectrum is concentrated in the interval $[-2,2]$. We prove a concentration bound for $N_I = N_I(W_n)$, the number of eigenvalues of ... More
Structure of large incomplete sets in abelian groupsOct 04 2006Let $G$ be a finite abelian group and $A$ be a subset of $G$. We say that $A$ is complete if every element of $G$ can be represented as a sum of different elements of $A$. In this paper, we study the following question: {\it What is the structure of a ... More
Sum-avoiding sets in groupsMar 09 2016May 29 2016Let $A$ be a finite subset of an arbitrary additive group $G$, and let $\phi(A)$ denote the cardinality of the largest subset $B$ in $A$ that is sum-avoiding in $A$ (that is to say, $b_1+b_2 \not \in A$ for all distinct $b_1,b_2 \in B$). The question ... More
The spectrum of random kernel matrices: universality results for rough and varying kernelsJun 17 2012May 21 2013We consider random matrices whose entries are f(<Xi,Xj>) or f(||Xi-Xj||^2) for iid vectors Xi in R^p with normalized distribution. Assuming that f is sufficiently smooth and the distribution of Xi's is sufficiently nice, El Karoui  showed that the ... More
Inverse Littlewood-Offord theorems and the condition number of random discrete matricesNov 08 2005Jan 28 2007Consider a random sum $\eta_1 v_1 + ... + \eta_n v_n$, where $\eta_1,...,\eta_n$ are i.i.d. random signs and $v_1,...,v_n$ are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as $\P(\eta_1 v_1 + ... + \eta_n v_n ... More Sumfree sets in groups: a surveyMar 09 2016Mar 15 2016We discuss several questions concerning sum-free sets in groups, raised by Erd\H{o}s in his survey "Extremal problems in number theory" (Proceedings of the Symp. Pure Math. VIII AMS) published in 1965. Among other things, we give a characterization for ... More Sum-avoiding sets in groupsMar 09 2016Jan 17 2017Let$A$be a finite subset of an arbitrary additive group$G$, and let$\phi(A)$denote the cardinality of the largest subset$B$in$A$that is sum-avoiding in$A$(that is to say,$b_1+b_2 \not \in A$for all distinct$b_1,b_2 \in B$). The question ... More Some Stability Properties of Parametric Quadratically Constrained Nonconvex Quadratic Programs in Hilbert SpacesJun 09 2017Stability of nonconvex quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces is investigated. We present several stability properties of the global solution map, and the continuity of the optimal value function, ... More Optimal Inverse Littlewood-Offord theoremsApr 22 2010Jan 16 2011Let eta_i be iid Bernoulli random variables, taking values -1,1 with probability 1/2. Given a multiset V of n integers v_1,..., v_n, we define the concentration probability as rho(V) := sup_{x} Pr(v_1 eta_1+...+ v_n eta_n=x). A classical result of Littlewood-Offord ... More John-type theorems for generalized arithmetic progressions and iterated sumsetsDec 30 2006May 21 2008A classical theorem of Fritz John allows one to describe a convex body, up to constants, as an ellipsoid. In this article we establish similar descriptions for generalized (i.e. multidimensional) arithmetic progressions in terms of proper (i.e. collision-free) ... More The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and FürediFeb 26 2010Apr 04 2011We give a new bound on the probability that the random sum$\xi_1 v_1 +...+ \xi_n v_n$belongs to a ball of fixed radius, where the$\xi_i$are iid Bernoulli random variables and the$v_i$are vectors in$\R^d$. As an application, we prove a conjecture ... More Local universality of zeroes of random polynomialsJul 16 2013Apr 30 2014In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials$f =\sum_{i=1}^n c_i \xi_i z^i$and$\tilde ... More
The Capacity of MIMO Channels with Per-Antenna Power ConstraintJun 24 2011We establish the optimal input signaling and the capacity of MIMO channels under per-antenna power constraint. While admitting a linear eigenbeam structure, the optimal input is no longer diagonalizable by the channel right singular vectors as with sum ... More
Studies of inflation and dark energy with coupled scalar fieldsFeb 03 2015Currently there is no definitive description for the accelerated expansion of the Universe at both early and late times; we know these two periods as the epochs of inflation and dark energy. Contained within this Thesis are two studies of inflation and ... More
Random matrices: Law of the determinantDec 04 2011Jan 13 2014Let $A_n$ 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 ... More
Thermodynamic uncertainty relation for time-delayed Langevin systemsSep 18 2018Feb 20 2019The thermodynamic uncertainty relation, which establishes a universal trade-off between the relative fluctuation of arbitrary currents and the dissipation, has been found for various Markovian systems. However, this relation has not been revealed for ... More
Generalized thermodynamic uncertainty relation via the fluctuation theoremFeb 18 2019Apr 14 2019The fluctuation theorem is the fundamental equality in nonequilibrium thermodynamics that is used to derive many important thermodynamic relations, such as the second law of thermodynamics and the Jarzynski equality. Recently, the thermodynamic uncertainty ... More
$N_6$ property for third Veronese embeddingsMar 22 2013The rational homology groups of the matching complexes are closely related to the syzygies of the Veronese embeddings. In this paper we will prove the vanishing of certain rational homology groups of matching complexes, thus proving that the third Veronese ... More
Universality of local eigenvalue statistics in random matrices with external sourceAug 05 2013Mar 18 2014Consider a random matrix of the form $W_n = M_n + D_n$, where $M_n$ is a Wigner matrix and $D_n$ is a real deterministic diagonal matrix ($D_n$ is commonly referred to as an external source in the mathematical physics literature). We study the universality ... More
Generalized thermodynamic uncertainty relation via fluctuation theoremFeb 18 2019The fluctuation theorem is the fundamental equality in nonequilibrium thermodynamics. It derives many important thermodynamic relations, such as the second law of thermodynamics and the Jarzynski equality. Recently, the thermodynamic uncertainty relation ... More
Uncertainty relations for time-delayed Langevin systemsFeb 19 2019Mar 11 2019The thermodynamic uncertainty relation, which establishes a universal trade-off between nonequilibrium current fluctuations and dissipation, has been found for various Markovian systems. However, this relation has not been revealed for non-Markovian systems; ... More
Uncertainty relation in the presence of information measurement and feedback controlApr 08 2019Apr 23 2019This study investigates the uncertainty of dynamical observables for classical systems manipulated by repeated measurements and feedback control. The precision of observables is expected to be enhanced in the presence of an external controller, but still ... More
Uncertainty Relations for Underdamped Langevin DynamicsJan 17 2019Mar 11 2019A universal trade-off between the precision of arbitrary currents and the dissipation cost, known as the thermodynamic uncertainty relation, has been investigated for various Markovian systems. Here, we study the thermodynamic uncertainty relation for ... More
Uncertainty relations in stochastic processes: An information inequality approachSep 10 2018Jan 07 2019The thermodynamic uncertainty relation is an inequality stating that it is impossible to attain higher precision than the bound defined by entropy production. In statistical inference theory, information inequalities assert that it is infeasible for any ... More
An algebraic method to calculate parameter regions for constrained steady-state distribution in stochastic reaction networksFeb 26 2018Jul 09 2018Steady state is an essential concept in reaction networks. Its stability reflects fundamental characteristics of several biological phenomena such as cellular signal transduction and gene expression. Because biochemical reactions occur at the cellular ... More
On the concentration of eigenvalues of random symmetric matricesSep 21 2000We prove that few largest (and most important) eigenvalues of random symmetric matrices of various kinds are very strongly concentrated. This strong concentration enables us to compute the means of these eigenvalues with high precision. Our approach uses ... More
Concentration of random determinants and permanent estimatorsMay 12 2009We show that the absolute value of the determinant of a matrix with random independent (but not necessarily iid) entries is strongly concentrated around its mean. As an application, we show that the Godsil-Gutman and Barvinok estimators for the permanent ... More
Diffusion-dynamics laws in stochastic reaction networksMay 10 2018Jan 17 2019Many biological activities are induced by cellular chemical reactions of diffusing reactants. The dynamics of such systems can be captured by stochastic reaction networks. A recent numerical study has shown that diffusion can significantly enhance the ... More
Circular law for random discrete matrices of given row sumMar 27 2012Let $M_n$ be a random matrix of size $n\times n$ and let $\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\mu_{M_n}$ of $M_n$ is defined as $$\mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le s; \Im(\lambda_k)\le ... More A characterization of incomplete sequences in F_p^dDec 04 2011A 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, ... More On the Rank of Random Sparse MatricesNov 16 2007We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to ... More Uncertainty relations for time-delayed Langevin systemsFeb 19 2019The thermodynamic uncertainty relation, which establishes a universal trade-off between the nonequilibrium current fluctuations and the dissipation, has been found for various Markovian systems. However, this relation has not been revealed for non-Markovian ... More Uncertainty relation in the presence of information measurement and feedback controlApr 08 2019Thermodynamic uncertainty relation, which provides a universal bound for relative fluctuation of arbitrary currents in nonequilibrium systems, has been developed for various systems. Here we study the uncertainty of dynamical observables for classical ... More The Rank of Random GraphsJun 17 2006We show that almost surely the rank of the adjacency matrix of the Erd\"os-R\'enyi random graph G(n,p) equals the number of non-isolated vertices for any c\ln n/n<p<1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the ... More Random matrix products: Universality and least singular valuesFeb 08 2018We establish local universality of the k-point correlation functions associated with products of independent iid random matrices, as the sizes of the matrices tend to infinity, under a moment matching hypothesis. We also prove Gaussian limits for the ... More Sparse random graphs: Eigenvalues and EigenvectorsNov 30 2010In this paper we prove the semi-circular law for the eigenvalues of regular random graph G_{n,d} in the case d\rightarrow \infty, complementing a previous result of McKay for fixed d. We also obtain a upper bound on the infinity norm of eigenvectors ... More Random matrices: tail bounds for gaps between eigenvaluesApr 01 2015May 02 2015Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices ... More Law of Iterated Logarithm for random graphsJul 29 2016Oct 10 2017A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables \{t_i\}_{i=1}^{\infty} with mean 0 and variance 1$$ \Pr \left[ ... More
Roots of random polynomials with coefficients having polynomial growthJul 17 2015Nov 19 2017In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coeffcients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even ... More
Iterates of holomorphic self-maps on pseudoconvex domains of finite and infinite type in $\mathbb C^n$Jul 16 2015We prove here the Wolff-Denjoy-type theorem for a very large class of pseudoconvex domains in $\mathbb C^n$ that may contain many classes of pseudoconvex domains of finite type and infinite type.
Almost periodic solutions of periodic linear partial functional differential equationsJul 10 2018We study conditions for the abstract periodic linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy ... More
Random perturbation and matrix sparsification and completionMar 02 2018We discuss general perturbation inequalities when the perturbation is random. As applications, we obtain several new results concerning two important problems: matrix sparsification and matrix completion.
On the Solution Existence of Nonconvex Quadratic Programming Problems in Hilbert SpacesApr 17 2016Apr 29 2016In this paper, we consider the quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces. By using the Legendre property of quadratic forms or the compactness of operators in the presentations of quadratic forms, ... More
Random perturbation of low rank matrices: Improving classical boundsNov 12 2013May 13 2016Matrix perturbation inequalities, such as Weyl's theorem (concerning the singular values) and the Davis-Kahan theorem (concerning the singular vectors), play essential roles in quantitative science; in particular, these bounds have found application in ... More
Random symmetric matrices are almost surely non-singularMay 09 2005Let $Q_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that $Q_n$ is non-singular with probability $1-O(n^{-1/8+\delta})$ for ... More
Random matrix products: Universality and least singular valuesFeb 08 2018Jul 13 2018We establish, under a moment matching hypothesis, the local universality of the correlation functions associated with products of $M$ independent iid random matrices, as $M$ is fixed, and the sizes of the matrices tend to infinity. This generalizes an ... More
Anti-concentration for polynomials of independent random variablesJul 03 2015Aug 07 2015We prove anti-concentration results for polynomials of independent random variables with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates. We discuss applications ... More
Spectra of lifted Ramanujan graphsNov 21 2009A random $n$-lift of a base graph $G$ is its cover graph $H$ on the vertices $[n]\times V(G)$, where for each edge $u v$ in $G$ there is an independent uniform bijection $\pi$, and $H$ has all edges of the form $(i,u),(\pi(i),v)$. A main motivation for ... More
Roots of random polynomials with arbitrary coefficientsJul 17 2015May 25 2016In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even ... More
Small ball probability, Inverse theorems, and applicationsDec 31 2012Let $\xi$ 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 $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability ... More
Random matrices: Law of the iterated logarithmMay 24 2016The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and many others. ... More
Electric gating induced bandgaps and enhanced Seebeck effect in zigzag bilayer graphene ribbonsFeb 02 2016We theoretically investigate effect of a transverse electric field generated by side gates and a vertical electric field generated by top, back gates on energy bands and transport properties of zigzag bilayer graphene ribbons (Bernal stacking). Using ... More
Normal vector of a random hyperplaneApr 17 2016Let 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 ... More
Law of Iterated Logarithm for random graphsJul 29 2016Sep 13 2016A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables $\{t_i\}_{i=1}^{\infty}$ with mean $0$ and variance $1$ $$\Pr \left[ ... More Random matrices: Universality of ESDs and the circular lawJul 30 2008Apr 23 2009Given an n \times n complex matrix A, let$$\mu_{A}(x,y):= \frac{1}{n} |\{1\le i \le n, \Re \lambda_i \le x, \Im \lambda_i \le y\}| be the empirical spectral distribution (ESD) of its eigenvalues $\lambda_i \in \BBC, i=1, ... n$. We consider the ... More
Real roots of random polynomials: expectation and repulsionSep 15 2014Let $P_{n}(x)= \sum_{i=0}^n \xi_i x^i$ be a Kac random polynomial where the coefficients $\xi_i$ are iid copies of a given random variable $\xi$. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal ... More
Stochastic Block Model and Community Detection in the Sparse Graphs: A spectral algorithm with optimal rate of recoveryJan 20 2015Jun 24 2015In this paper, we present and analyze a simple and robust spectral algorithm for the stochastic block model with $k$ blocks, for any $k$ fixed. Our algorithm works with graphs having constant edge density, under an optimal condition on the gap between ... More
The Non-Adiabatic Pressure Perturbation and Non-Canonical Kinetic Terms in Multifield InflationNov 20 2012Nov 30 2012The evolution of the non-adiabatic pressure perturbation during inflation driven by two scalar fields is studied numerically for three different types of models. In the first model, the fields have standard kinetic terms. The other two models considered ... More