Van Vu

• Professor (Full) at Yale University

We consider random orthonormal polynomials F n ( x ) = ∑ i = 0 n ξ i p i ( x ) , \begin{equation*} F_{n}(x)=\sum _{i=0}^{n}\xi _{i}p_{i}(x), \end{equation*} where ξ 0 \xi _{0} , …, ξ n \xi _{n} are independent random variables with zero mean, unit variance and uniformly bounded ( 2 + ε ) (2+\varepsilon ) moments, and ( p n ) n = 0 ∞ (p_n)_{n=0}^{\i...

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 re...

The 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 many years, and, based on numerical data, have raised a number of questions about the distribution...

In 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 when the coefficients are not explicit. Our results also hold for series; in particular, we pro...

A 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 with mean 0 and variance 1 In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number...

The Davis-Kahan-Wedin $\sin \Theta$ theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis-Kahan-Wedin $\sin \Theta$ theorem when the perturbation is a Gaussian random matrix. Under cert...

Let $M_n$ 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 constant $\delta > 0$ and $\xi_{ij}$ are i.i.d. centered, subgaussian random variables. We show th...

Let $M_n$ 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 constant $\delta > 0$ and $\xi_{ij}$ are i.i.d. centered, subgaussian random variables. We show th...

We 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 centered linear spectral statistics of products of independent GinUE matrices, which we then exte...

We 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 earlier result of Tao and the third author for the case $M=1$. We also prove Gaussian limits for t...

In this paper, we study 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 of Kac and Littlewood-Offord in the 1940s, random functions of this type have been stu...

In 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 $$2^{-\left(0.5+o(1)\right)n^2} \le P\left(M_n \text{ is normal}\right) \le 2^{-(0.302+o(1))n^{2...

In 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 $$2^{-\left(0.5+o(1)\right)n^2} \le P\left(M_n \text{ is normal}\right) \le 2^{-(0.302+o(1))n^{2...

Matrix 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 data analysis as well as related areas of engineering and computer science. In many situations, t...

We 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 correlation functions of the real roots of P_n and a comparison argument exploiting local laws and repu...

We 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 correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws...

We show that several statistics of the number of intersections between random eigenfunctions of general eigenvalues with a given smooth curve in flat tori are universal under various families of randomness.

We show that several statistics of the number of intersections between random eigenfunctions of general eigenvalues with a given smooth curve in flat tori are universal under various families of randomness.

In this short survey, we discuss the notion of anti-concentration and describe various ideas used to obtain anti-concentration inequalities, together with several open questions.

Gaps (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 with discrete entries and the first super-polynomial bound on the probability that a random grap...

A 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[ \limsup_{n\rightarrow \infty} \frac{ \sum_{i=1}^n t_i }{\sigma_n \sqrt {2 \log \log n }} =1 \...

We 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 hypergraph $H^{k}_{n,p}$ contains $N:=(1-o(1))\binom{n-1}{k-1}p$ edge-disjoint perfect matching...

We 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 hypergraph $H^{k}_{n,p}$ contains $N:=(1-o(1))\binom{n-1}{k-1}p$ edge-disjoint perfect matching...

The 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. In this notes, we discuss the following problem: Is it possible to prove the law of the iterated l...

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 rand...

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 rand...

Sum-avoiding sets in groups, Discrete Analysis 2016:15, 27 pp. Let $A$ be a subset of an Abelian group $G$. A subset $B\subset A$ is called _sum-avoiding in $A$_ if no two elements of $B$ add up to an element of $A$. Write $\phi(A)$ for the size of the largest sum-avoiding subset of $A$. If $G=\mathbb Z$ and $|A|=n$, then it is known that $\phi(A)...

We 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 large sets $A$ in an abelian group $G$ which do not contain a subset $B$ of fixed size $k$ such t...

Eigenvectors 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.

In 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 when the coefficients are not explicit. Our results also hold for series; in particular, we pro...

We prove anti-concentration results for polynomials of independent Rademacher random variables, with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates. As an application, we address a challenge in complexity theory posed by Razborov and Viola.

As an extension of Polya’s classical result on random walks on the square grids (\({\mathbf {Z}}^d\)), 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 after n steps is at most \(n^{-d/2 - d/(d-2) +o(1) }\), which is sharp. The re...

In 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. In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sh...

Let $A$ 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 circumstances, it is clear that this problem is underdetermined. However, in the case when $X$ is...

In 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 the density inside a block and the density between the blocks. As a co-product, we settle an open...

Let $M_n = (\xi_{ij})_{1 \leq i,j \leq n}$ 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. As a corollary, we deduce that the Erd{\H o}s-Renyi random graph has simple spectrum asympt...

As 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 after $n$ steps is at most $n^{-d/2 - d/(d-2) +o(1)}$, which is sharp. The real surprise is...

Let $P_{n}(x)= \sum _{i=0}^n \xi _i x^i$ be a Kac random polynomial where the coefficients $\xi _i$ are i.i.d. copies of a given random variable $\xi $. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a real double root. As an application, we consider...

In this paper, we present a simple, yet useful, concentration result concerning random (weighted) projections in high dimensional spaces. As one application, we prove a general concentration result for random quadratic forms, which extended a classical result of Hanson and Wright and improved several recent results. In another application, we show...

Finding a hidden partition in a random environment is a general and important problem which contains as subproblems many important questions, such as finding a hidden clique, finding a hidden colouring, finding a hidden bipartition, etc . In this paper we provide a simple SVD algorithm for this purpose, addressing a question of McSherry. This algor...

For 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 of the circular law, regardless of the joint distribution of the mirror entries in each matrix....

Roots 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 of real roots is $\frac{2}{\pi} \log n + o(\log n)$. In this paper, we determine the true natu...

Matrix 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 data analysis as well as related areas of engineering and computer science. In many situations, t...

Consider 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 of the local eigenvalue statistics of W_n for a general class of Wigner matrices M_n and diagonal mat...

In 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 $\tilde {f} =\sum _{i=1}^n c_i \tilde {\xi }_i z^i$, where the ξi and $\tilde {\xi }_i$ are iid rand...

In 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 of Erd\H{o}s-R\'enyi random graph $G(n,p)$, answering a question raised by Dekel-Lee-Linial. C...

Let $\xi$ 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 and its applications. We discuss the small ball problem, the aim of which is to estimate the maxi...

We 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 spectral distributions of these matrices behave as if f is linear in the Marchenko--Pastur limit....

It 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 $\C$ with kernel $K_\infty(z,w) := \frac{1}{\pi} e^{-|z|^2/2 - |w|^2/2 + z \bar{w}}$ in the limi...

Let $M_n$ 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 $$\mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le s; \Im(\lambda_k)\le t\}.$$ The circular law theorem in random matrix theory asserts that if the entries of $...

A sequence A of elements an additive group G is incomplete if there exists a group element that cannot 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...

In 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 universality of the sine kernel and the Central limit theorem of several spectral parameters. We...

Let $W_n= \frac{1}{\sqrt n} M_n$ 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 $W_n$ in an interval $I$. Our result shows that $N_I$ decays exponentially with standard deviation...

We 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.

Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero and variance one. We show that the logarithm of $|det A_n|$ satisfies a central limit theorem. More precisely, $$\sup_{x\in R} |P(\frac{\log (|det A_n|)- 1/2 \log(n-1)!}{\sqrt{1/2 \log n}}\le x) -\Phi(x)| \le \log^{-1/3 +o(1)} n.$$

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 grou...

Computing 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, an exact calculation is hard to achieve, and the following problem is of importance: How much does a small perturbation to the matrix change the singular vectors? Answering this question...

We 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 distributed like $N(\log \sqrt{n!} - 1/2 \log n, 1/2 \log n)_\R$ when one matches moments w...

The 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 of the matrix. In this paper, we extend the four moment theorem to also cover the coefficients of...

This is a continuation of our earlier paper [25] on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in [25] from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov [23] for the largest eigenvalues, assum...

A 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 correlation function of the Dyson sine process in the asymptotic limit $n \to \infty$. The...

The 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, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Mom...

Consider 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 that $\lambda_i(M_n)$ concentrates around $\gamma_i \sqrt n$, where $\int_{-\infty}^{\gamma_i} \r...

Let 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 and Erdos from the 1940s asserts that if the v_i are non-zero, then rho(V) is O(n^{-...

We give a new bound on the probability that the random sum ξ 1v 1+…+ξ n v n belongs to a ball of fixed radius, where the ξ i are i.i.d. Bernoulli random variables and the v i are vectors in R d . As an application, we prove a conjecture of Frankl and Füredi (raised in 1988), which can be seen as the high dimensional version of the classical Littlew...

Let n be a large integer and Mn be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that Mn is singular. For a constant 0<p<1 and a constant positive integer r, we will define a property p-...

This is the first part of a series of surveys on random matrices. In this part, we focus on problems and results of combinatorial nature.

We 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 for some sufficiently large constant $C_0$. The main result of this paper is a Four Moment the...

A random n-lift of a base-graph G is its cover graph H on the vertices [n]×V(G), where for each edge uv in G there is an independent uniform bijection π, and H has all edges of the form (i,u),(π(i),v). A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan.Let G b...

A 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 studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a gra...

Text: Let f (n, m) be the cardinality of largest subset of {1, 2, ..., n} which does not contain a subset whose elements sum to m. In this note, we show thatf (n, m) = (1 + o (1)) frac(n, snd (m)) for all n (log n)1 + ε{lunate} ≤ m ≤ frac(n2, 9 log2 n), where snd (m) is the smallest integer that does not divide m. This proves a conjecture of Alon p...

We show that if 𝒜 is a finite set of d × d well-conditioned matrices with complex entries, then the following sum–product estimate holds | 𝒜 + 𝒜 | × |𝒜·𝒜| = Ω (|𝒜| 5/2).

This 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, we prove a variant of the universality results of Soshnikov for the largest eigenvalues, assuming...

The famous Erdős-Heilbronn conjecture (first proved by Dias da Silva and Hamidoune in 1994) asserts that if A is a subset of Z/pZ, the cyclic group of the integers modulo a prime p, then |A ̂+ A | � min{2 |A | − 3,p}. The bound is sharp, as is shown by choosing A to be an arithmetic progression. A natural inverse result was proven by Karolyi in 200...

The famous Erdős-Heilbronn conjecture (first proved by Dias da Silva and Hamidoune in 1994) asserts that if $A$ is a subset of ${\Bbb Z}/p{\Bbb Z}$, the cyclic group of the integers modulo a prime $p$, then $|A\widehat{+}A| \ge \min\{2|A| -3,p\}. $ The bound is sharp, as is shown by choosing $A$ to be an arithmetic progression. A natural inverse re...

In this paper, we consider the ensemble of $n \times n$ Wigner Hermitian matrices $H = (h_{\ell k})_{1 \leq \ell,k \leq n}$ that generalize the Gaussian unitary ensemble (GUE). The matrix elements $h_{k\ell} = \bar h_{ \ell k}$ are given by $h_{\ell k} = n^{-1/2} ( x_{\ell k} + \sqrt{-1} y_{\ell k} )$, where $x_{\ell k}, y_{\ell k}$ for $1 \leq \el...

We 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 of a strictly positive matrix give sub-exponential approximation ratios with high probability.

Let $M_n$ 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 of $(3/4 + o(1))^n$ proven by Tao and Vu in arXiv:math/0501313v2. This paper follows a similar ap...

Let Zp be the finite field of prime order p and A be a subsequence of Zp. 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 every element of Zp as a sum of some elements of A?(3) When can one represent every element of Zp as a sum of l elem...

Let $\a$ 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 $M_n(\a)$. ($\sigma_n(M_n(\a))^2$ is also usually interpreted as the least eigenvalue of the Wishart matrix $M_n M_n^{\ast}$.) We s...

Let ηi,i = 1,…,n be iid Bernoulli random variables. Given a multiset vof n numbers v1,…,vn, the concentration probability P1(v) of v is defined as P1(v) := supxP(v1η1+ …vnηn = x). A classical result of Littlewood–Offord and Erdős from the 1940s asserts that if the vi are nonzero, then this probability is at most O(n-1/2). Since then, many researche...

In 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 challenging problems. Here we focus on random and pseudo-random graphs and prove several sharp resu...

The 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 to the uniform distribution on the unit disk $\{z \in \C: |z| \leq 1 \}$. After a long sequen...

A 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 that this maximum cardinality is of order $n^{1/3+o(1)}$. Comment: 33 pages

In this survey, we discuss some basic problems concerning random matrices with discrete distributions. Several new results, tools and conjectures will be presented.

We 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 giant component (a.s.) has full rank in this range.

A 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) generalized arithmetic progressions, in both torsion-free and torsion settings. We also o...

Given 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 limiting distribution (both in probability and in the almost sure convergence sense) of the normal...

Let H be a fixed graph on v vertices. For an n-vertex graph G with n divisible by v, an H-factor of G is a collection of n/v copies of H whose vertex sets partition V (G). In this work, we consider the threshold thH(n) of the property that an Erdős-Rényi random graph (on n points) contains an H-factor. Our results determine thH(n) for all strictly...

We 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 as number theory, combinatorics and mathematical physics.

A few years ago, Spielman and Teng initiated the study of Smooth analysis of the condition number and the least singular value of a matrix. Let x be a complex variable with mean zero and bounded variance. Let N n be the random matrix of sie n whose entries are iid copies of x and M a deterministic matrix of the same size. The goal of smooth analysi...

We show that the permanent of an n×n matrix with iid Bernoulli entries ±1 is of magnitude with probability 1−o(1). In particular, it is almost surely non-zero.

We show that the permanent of an $n \times n$ matrix with iid Bernoulli entries $\pm 1$ is of magnitude $n^{({1/2}+o(1))n}$ with probability $1-o(1)$. In particular, it is almost surely non-zero.

For convex bodies K with boundary in ℝ d , we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem.

We show that the number of distinct distances in a set of n points in ℝ d is Ω(n 2/d − 2 / d(d + 2)), d ≥ 3. Erdős’ conjecture is Ω(n 2/d ).