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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

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 $F_p^d$Dec 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

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

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

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

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

Classification of 5-Dimensional MD-Algebras Having Non-Commutative Derived IdealsJan 11 2011Feb 09 2012The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e. five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbits) are orbits of zero or maximal dimension. The main ... 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

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: 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

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

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 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

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

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

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

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 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

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 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: 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

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

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 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

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

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

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

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 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

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

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

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

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

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 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

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

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

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 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 [17] showed that the ... 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

$N_6$ 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

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

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

Modulation of bandgap in bilayer armchair graphene ribbons by tuning vertical and transverse electric fieldsNov 22 2016We investigate the effects of external electric fields on the electronic properties of bilayer armchair graphene nano-ribbons. Using atomistic simulations with Tight Binding calculations and the Non-equilibrium Green function formalism, we demonstrate ... 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

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

Singular solutions with vorticity control for a nonlocal system of evolution equationsOct 27 2016Oct 28 2016We investigate a system of nonlocal transport equations in one spatial dimension. The system can be regarded as a model for the 3D Euler equations in the hyperbolic flow scenario. We construct blowup solutions with control up to the blowup time.

On semi-global invariants for focus-focus singularitiesAug 30 2002This article gives a classification, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a 4-dimensional symplectic manifold, in a full neighbourhood of a singular leaf of focus-focus (=nodal) type. ... More

Moment polytopes for symplectic manifolds with monodromyApr 08 2005A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such ``almost-toric 4-manifolds'' which admits a Hamiltonian $S^1$-action we show that one can associate ... 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.

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

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

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

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

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

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

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 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

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

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

Positive solutions of an integral equation and Riemann hypothesisMar 25 2010May 30 2016We prove that if an integral equation has a positive solution then all complex roots of the famous Riemann zeta function are distinct and having the real part 1/2. We also prove that the minimal distance between two consecutive real simple roots of the ... More

The higher order $q$-Dolan-Grady relations and quantum integrable systemsJan 23 2016Jan 31 2016In this thesis, the connection between recently introduced algebraic structures (tridiagonal algebra, $q$-Onsager algebra, generalized $q-$Onsager algebras), related representation theory (tridiagonal pair, Leonard pair, orthogonal polynomials), some ... More

Long arithmetic progressions in sumsets: Thresholds and BoundsJul 26 2005Aug 11 2005For a set $A$ of integers, the sumset $lA =A+...+A$ consists of those numbers which can be represented as a sum of $l$ elements of $A$ $$lA =\{a_1+... a_l| a_i \in A_i \}. $$ A closely related and equally interesting notion is that of $l^{\ast}A$, which ... More

On a Subclass of 5-Dimensional Solvable Lie Algebras Which Have 3-Dimensional Commutative Derived IdealMar 03 2006The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e., five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbit) are orbit of zero or maximal dimension. The main results ... 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

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

Random Matrix Ensembles with Split Limiting BehaviorSep 11 2016Sep 15 2016We introduce a new family of $N\times N$ random real symmetric matrix ensembles, the $k$-checkerboard matrices, whose limiting spectral measure has two components which can be determined explicitly. All but $k$ eigenvalues are in the bulk, and their behavior, ... 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

Eigenvectors of random matrices: A surveyJan 14 2016Jun 13 2016Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of eigenvectors when the matrix (or graph) is random.

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

On the singularity probability of discrete random matricesMay 04 2009Let $M_n$ be an $n$ by $n$ random matrix where each entry is +1 or -1 independently with probability 1/2. Our main result implies that the probability that $M_n$ is singular is at most $(1/\sqrt{2} + o(1))^n$, improving on the previous best upper bound ... More

On the number of real roots of random polynomialsFeb 19 2014Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the expectation of the number ... More

Spectral asymptotics via the semiclassical Birkhoff normal formMay 03 2006This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential ... More

Powers of sums and their homological invariantsJul 25 2016Jul 27 2016Let $R$ and $S$ be standard graded algebras over a field $k$, and $I \subseteq R$ and $J \subseteq S$ homogeneous ideals. Denote by $P$ the sum of the extensions of $I$ and $J$ to $R\otimes_k S$. We investigate several important homological invariants ... More

On-Demand Routing Algorithm with Mobility Prediction in the Mobile Ad-hoc NetworksSep 26 2016In this paper, we propose an ad-hoc on-demand distance vector routing algorithm for mobile ad-hoc networks taking into account node mobility. Changeable topology of such mobile ad-hoc networks provokes overhead messages in order to search available routes ... More

Non-separability and steerability of two-qubit states from the geometry of steering outcomesApr 01 2016Jul 27 2016When two qubits A and B are in an appropriate state, Alice can remotely steer Bob's system B into different ensembles by making different measurements on A. This famous phenomenon is known as quantum steering, or Einstein-Podolsky-Rosen steering. Importantly, ... More

Regularity over homomorphisms and a Frobenius characterization of Koszul algebrasMar 21 2013Feb 02 2015Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $\phi:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^{\phi}\!M$ be the abelian group $M$ with the $R$-module ... More

Integrable systems, symmetries and quantizationApr 21 2017Oct 16 2017These notes correspond to a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularity and expose recent research in connection ... More

Constraints on Disformal Couplings from the Properties of the Cosmic Microwave Background RadiationMar 07 2013Aug 29 2013Certain modified gravity theories predict the existence of an additional, non-conformally coupled scalar field. A disformal coupling of the field to the Cosmic Microwave Background (CMB) is shown to affect the evolution of the energy density in the radiation ... More

Effect of transitions in the Planck mass during inflation on primordial power spectraJun 20 2014Oct 23 2014We study the effect of sudden transitions in the effective Planck mass during inflation on primordial power spectra. Specifically, we consider models in which this variation results from the non-minimal coupling of a Brans-Dicke type scalar field. We ... More

Mapping IncidencesNov 28 2007Apr 15 2011We show that any finite set S in a characteristic zero integral domain can be mapped to the finite field of order p, for infinitely many primes p, preserving all algebraic incidences in S. This can be seen as a generalization of the well-known Freiman ... More

Scalable Support Vector Machine for Semi-supervised LearningJun 22 2016Sep 06 2016Owing to the prevalence of unlabeled data, semisupervised learning has been one of the most prominent machine learning paradigms, and applied successfully in many real-world applications. However, most of existing semi-supervised learning methods are ... More

Products of independent elliptic random matricesMar 24 2014Mar 02 2015For fixed $m > 1$, we study the product of $m$ independent $N \times N$ elliptic random matrices as $N$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $1$, to the $m$-th power ... More

Using the Johnson-Lindenstrauss lemma in linear and integer programmingJul 03 2015The Johnson-Lindenstrauss lemma allows dimension reduction on real vectors with low distortion on their pairwise Euclidean distances. This result is often used in algorithms such as $k$-means or $k$ nearest neighbours since they only use Euclidean distances, ... More

Isospectrality for quantum toric integrable systemsNov 25 2011We settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of such a system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable ... More

Singular Bohr-Sommerfeld Rules for 2D Integrable SystemsMay 26 2000In this paper, we describe Bohr-Sommerfeld rules for semi-classical completely integrable systems with 2 degrees of freedom with non degenerate singularities (Morse-Bott singularities) under the assumption that the energy level of the first Hamiltonian ... More

K-Theory for the Leaf Space of Foliations formed by the Generic K-Ornits of some indecomposable $MD_5$-GroupsMar 27 2010Jan 11 2011The paper is a continuation of the authors' work in which we considered foliations formed by the maximal dimensional K-orbits ($MD_5$-foliations) of connected $MD_5$-groups such that their Lie algebras have 4-dimensional commutative derived ideals and ... More

The Topology of Foliations Formed by the Generic K-Orbits of a Subclass of the Indecomposable MD5-GroupsJan 18 2008Feb 23 2008The present paper is a continuation of [13], [14] of the authors. Specifically, the paper considers the MD5-foliations associated to connected and simply connected MD5-groups such that their Lie algebras have 4-dimensional commutative derived ideal. In ... More