Search

We consider the cokernel $G_n = \mathbf{Cok}(A_{k} \cdots A_2 A_1)$ of a product of independent $n \times n$ random integer matrices with iid entries from generic nondegenerate distributions, in the regime where both $n$ and $k$ are sent to $\infty$ simultaneously. In this regime we show that the cokernel statistics converge universally to the reflecting Poisson sea, an interacting particle system constructed in arXiv:2312.11702, at the level of $1$-point marginals. In particular, $\operatorname{corank}(A_{k} \cdots A_2 A_1 \pmod{p}) \sim \log_p k$, and its fluctuations are $O(1)$ and converge to a discrete random variable defined in arXiv:2310.12275. The main difference with previous works studying cokernels of random matrices is that $G_n$ does not converge to a random finite group; for instance, the $p$-rank of $G_n$ diverges. This means that the usual moment method for random groups does not apply. Instead, we proceed by proving a `rescaled moment method' theorem applicable to a general sequence of random groups of growing size. This result establishes that fluctuations of $p$-ranks and other statistics still converge to limit random variables, provided that certain rescaled moments $\mathbb{E}[\#\operatorname{Hom}(G_n,H)]/C(n,H)$ converge., Comment: 35 pages. First version, comments welcome!

Many statistics of roots of random polynomials have been studied in the literature, but not much is known on the concentration aspect. In this note we present a systematic study of this question, aiming towards nearly optimal bounds to some extent. Our method is elementary and works well for many models of random polynomials, with gaussian or non-gaussian coefficients., Comment: arXiv admin note: text overlap with arXiv:2008.04111, arXiv:1912.12051

The Kac polynomial $$f_n(x) = \sum_{i=0}^{n} \xi_i x^i$$ with independent coefficients of variance 1 is one of the most studied models of random polynomials. It is well-known that the empirical measure of the roots converges to the uniform measure on the unit disk. On the other hand, at any point on the unit disk, there is a hole in which there are no roots, with high probability. In a beautiful work \cite{michelen2020real}, Michelen showed that the holes at $\pm 1$ are of order $1/n$. We show that in fact, all the hole radii are of the same order. The same phenomenon is established for the derivatives of the Kac polynomial as well., Comment: 23 pages, 4 figures

In this note we show that the singular probability of the adjacency matrix of a random $d$-regular graph on $n$ vertices, where $d$ is fixed and $n \to \infty$, is bounded by $n^{-1/3+o(1)}$. This improves a recent bound by Huang. Our method is based on the study of the singularity problem modulo a prime together with an inverse-type result on the decay of the characteristic function. The latter is related to the inverse Kneser's problem in combinatorics., Comment: 24 pages, 2 figures

We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and $\{p_n\}_{n=0}^{\infty}$ is the system of orthonormal polynomials with respect to a general exponential weight $W$ on the real line. This class of orthogonal polynomials includes the popular Hermite and Freud polynomials. We establish universality for the leading asymptotics of the expected number of real roots of $P_n$, both globally and locally. In addition, we find an almost sure limit of the measures counting all roots of $P_n.$ This is accomplished by introducing new ideas on applications of the inverse Littlewood-Offord theory in the context of the classical three term recurrence relation for orthogonal polynomials to establish anti-concentration properties, and by adapting the universality methods to the weighted random orthogonal polynomials of the form $W P_n.$

In this paper we study the cokernels of various random integral matrix models, including random symmetric, random skew-symmetric, and random Laplacian matrices. We provide a systematic method to establish universality under very general randomness assumption. Our highlights include both local and global universality of the cokernel statistics of all these models. In particular, we find the probability that a sandpile group of an Erdos-Renyi random graph is cyclic, answering a question of Lorenzini from 2008., Comment: 68 pages

For random integer matrices $M_1,\ldots,M_k \in \operatorname{Mat}_n(\mathbb{Z})$ with independent entries, we study the distribution of the cokernel $\operatorname{cok}(M_1 \cdots M_k)$ of their product. We show that this distribution converges to a universal one as $n \to \infty$ for a general class of matrix entry distributions, and more generally show universal limits for the joint distribution of $\operatorname{cok}(M_1),\operatorname{cok}(M_1M_2),\ldots,\operatorname{cok}(M_1 \cdots M_k)$. Furthermore, we characterize the universal distributions arising as marginals of a natural generalization of the Cohen-Lenstra measure to sequences of abelian groups with maps between them, which weights sequences inversely proportionally to their number of automorphisms. The proofs develop an extension of the moment method of Wood to joint moments of multiple groups, and rely also on the connection to Hall-Littlewood polynomials and symmetric function identities. As a corollary we obtain an explicit universal distribution for coranks of random matrix products over $\mathbb{F}_p$ as the matrix size tends to infinity., Comment: 48 pages, no figures. Comments welcome!

In this note we study the number of real roots of a wide class of random orthogonal polynomials with gaussian coefficients. Using the method of Wiener Chaos we show that the fluctuation in the bulk is asymptotically gaussian, even when the local correlations are not necessarily the same., Comment: 33 pages

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

We give a new proof of a recent resolution by Michelen and Sahasrabudhe of a conjecture of Shepp and Vanderbei that the moduli of roots of Gaussian Kac polynomials of degree $n$, centered at $1$ and rescaled by $n^2$, should form a Poisson point process. We use this new approach to verify a conjecture of Michelen and Sahasrabudhe that the Poisson statistics are in fact universal., Comment: 2nd version contains a correction of the extension in section 8

It has been shown in a recent work by Yakir-Zeitouni that the minimum modulus of random trigonometric polynomials with Gaussian coefficients has a limiting exponential distribution. We show this is a universal phenomenon. Our approach relates the joint distribution of small values of the polynomial at a fixed number $m$ of points on the circle to the distribution of a certain random walk in a $4m$-dimensional phase space. Under Diophantine approximation conditions on the angles, we obtain strong small ball estimates and a local central limit theorem for the distribution of the walk., Comment: 46 pages, 2 figures; Discrete Analysis, October 2021

We show that in two dimensional flat tori the number of intersections between random eigenfunctions of general eigenvalues and a given smooth curve is almost exponentially concentrated around its mean, even when the randomness is not gaussian., Comment: 14 pages, comments are welcome. arXiv admin note: text overlap with arXiv:1912.12051

We show that the number of real roots of random trigonometric polynomials with i.i.d. coefficients, which are either bounded or satisfy the logarithmic Sobolev inequality, satisfies an exponential concentration of measure., Comment: 17 pages

In this paper, we study the number of real roots of random trigonometric polynomials with iid coefficients. When the coefficients have zero mean, unit variance and some finite high moments, we show that the variance of the number of real roots is asymptotically linear in terms of the expectation; furthermore, the multiplicative constant in this linear relationship depends only on the kurtosis of the common distribution of the polynomial's coefficients. This result is in sharp contrast to the classical Kac polynomials whose corresponding variance depends only on the first two moments. Our result is perhaps the first paper to establish the variance for general distribution of the coefficients including discrete ones, for a model of random polynomials outside the family of the Kac polynomials. Our method gives a fine comparison framework throughout Edgeworth expansion, asymptotic Kac-Rice formula and a detailed analysis of characteristic functions., Comment: 2 figures

In this note we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank of random matrices over F_p and the equi-distribution behavior of normal vectors of random hyperplanes. We also study the probability that a random square matrix is eigenvalue-free, or when its characteristic polynomial is divisible by a given irreducible polynomial in the limit n to infinity in F_p. We show that these statistics are universal, extending results of Stong and Neumann-Praeger beyond the uniform model., Comment: 39 pages, typos corrected, title changed

We study the distribution of the cokernels of adjacency matrices (the Smith groups) of certain models of random $r$-regular graphs and directed graphs, using recent mixing results of M\'esz\'aros. We explain how convergence of such distributions to a limiting probability distribution implies asymptotic nonsingularity of the matrices, giving another perspective on recent results of Huang and M\'esz\'aros on asymptotic nonsingularity of adjacency matrices of random regular directed and undirected graphs, respectively. We also remark on the new distributions on finite abelian groups that arise, in particular in the $p$-group aspect when $p\mid r$.

For a random matrix of entries sampled independently from a fairly general distribution in Z we study the probability that the cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that the linear map between the integer lattices given by the matrix is surjective. We show that these statistics are asymptotically universal (as the size of the matrix goes to infinity), given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra, even when the distribution of matrix entries is very distorted. Our method is robust and works for Laplacians of random digraphs and sparse matrices with the probability of an entry non-zero only n^{-1+epsilon}., Comment: 44 pages

We show that a nearly square iid random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz. Our result extends to sparse matrices as well as to matrices of dependent entries., Comment: 20 pages

It is well-known that distances in random iid matrices are highly concentrated around their mean. In this note we extend this concentration phenomenon to Wigner matrices. Exponential bounds for the lower tail are also included., Comment: 33 pages, 1 figure; to appear in Linear Algebra and its Applications

We address overcrowding estimates for the singular values of random iid matrices, as well as for the eigenvalues of random Wigner matrices. We show evidence of long range separation under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds, Comment: 22 pages (replacing old title "Random matrices: repulsion in spectrum".), JFA (2018)

Suppose that A_1,\dots, A_N are independent random matrices whose atoms are iid copies of a random variable \xi of mean zero and variance one. It is known from the works of Newman et. al. in the late 80s that when \xi is gaussian then N^{-1} \log ||A_N \dots A_1|| converges to a non-random limit. We extend this result to more general matrices with explicit rate of convergence. Our method relies on a simple connection between structures and dynamics., Comment: 35 pages

Let v_1,...,v_{n-1} be n-1 independent vectors in R^n (or C^n). We study x, the unit normal vector of the hyperplane spanned by the v_i. Our main finding is that x resembles a random vector chosen uniformly from the unit sphere, under some randomness assumption on the v_i. Our result has applications in random matrix theory. Consider an n by n random matrix with iid entries. We first prove an exponential bound on the upper tail for the least singular value, improving the earlier linear bound by Rudelson and Vershynin. Next, we derive optimal delocalization for the eigenvectors corresponding to eigenvalues of small modulus., Comment: 24 pages, 2 figures. A portion of this work substantially improves a previous preprint of the first author

In this short note we address a gaussian property of normal vectors in random non-Hermitian matrices. The approach uses a simple geometric and comparison technique., Comment: This paper has been withdrawn by the author due to a more recent note written with Van H. Vu

We provide a characterization for anti-concentration of inhomogeneous random walks in non-abelian groups. In application we extend the classical bounds by Erdos-Littlewood-Offord and Sarkozy-Szemeredi to non-abelian settings., Comment: 27 pages, new applications added, typos and errors corrected

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 maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas., Comment: 47 pages

Let $X$ be a matrix sampled uniformly from the set of doubly stochastic matrices of size $n\times n$. We show that the empirical spectral distribution of the normalized matrix $\sqrt{n}(X-{\mathbf {E}}X)$ converges almost surely to the circular law. This confirms a conjecture of Chatterjee, Diaconis and Sly., Comment: Published in at http://dx.doi.org/10.1214/13-AOP877 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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 $M_n$ are i.i.d. copies of a random variable with mean zero and variance $\sigma^2$, then the empirical spectral distribution of the normalized matrix $\frac{1}{\sigma\sqrt{n}}M_n$ of $M_n$ converges almost surely to the uniform distribution $\mu_\cir$ over the unit disk as $n$ tends to infinity. In this paper we show that the empirical spectral distribution of the normalized matrix of $M_n$, a random matrix whose rows are independent random $(-1,1)$ vectors of given row-sum $s$ with some fixed integer $s$ satisfying $|s|\le (1-o(1))n$, also obeys the circular law. The key ingredient is a new polynomial estimate on the least singular value of $M_n$.

Let $\eta_i, i=1,..., n$ be iid Bernoulli random variables, taking values $\pm 1$ with probability 1/2. Given a multiset $V$ of $n$ elements $v_1, ..., v_n$ of an additive group $G$, we define the \emph{concentration probability} of $V$ as $$\rho(V) := \sup_{v\in G} P(\eta_1 v_1 + ... \eta_n v_n =v). $$ An old result of Erdos and Moser asserts that if $v_i $ are distinct real numbers then $\rho(V)$ is $O(n^{-3/2}\log n)$. This bound was then refined by Sarkozy and Szemeredi to $O(n^{-3/2})$, which is sharp up to a constant factor. The ultimate result dues to Stanley who used tools from algebraic geometry to give a complete description for sets having optimal concentration probability; the result now becomes classic in algebraic combinatorics. In this paper, we will prove that the optimal sets from Stanley's work are stable. More importantly, our result gives an almost complete description for sets having large concentration probability.

It is shown that a random $(0,1)$ matrix whose rows are independent random vectors of exactly $n/2$ zero components is non-singular with probability $1-O(n^{-C})$ for any $C>0$. The proof uses a non-standard inverse-type Littlewood-Offord result., Comment: 16 pages

A sequence $A$ of elements an additive group $G$ is {\it incomplete} if there exists a group element that {\it can not} be expressed as a sum of elements from $A$. The study of incomplete sequences is a popular topic in combinatorial number theory. However, the structure of incomplete sequences is still far from being understood, even in basic groups. The main goal of this paper is to give a characterization of incomplete sequences in the vector space $F_p^d$, where $d$ is a fixed integer and $p$ is a large prime. As an application, we give a new proof for a recent result by Gao-Ruzsa-Thangadurai on the Olson's constant of $\F_p^2$ and partially answer their conjecture concerning $F_p^3$., Comment: 14 pages

Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_n|$ satisfies a central limit theorem. More precisely, \begin{eqnarray*}\sup_{x\in {\mathbf {R}}}\biggl|{\mathbf {P}}\biggl(\frac{\log(|\det A_n|)-({1}/{2})\log (n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf {P}}\bigl(\mathbf {N}(0,1)\le x\bigr)\biggr|\\\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}, Comment: Published in at http://dx.doi.org/10.1214/12-AOP791 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Motivated by the inverse Littlewood-Offord problem for linear forms, we study the concentration of quadratic forms. We show that if this form concentrates on a small ball with high probability, then the coefficients can be approximated by a sum of additive and algebraic structures., Comment: 17 pages. This is the first part of http://arxiv.org/abs/1101.3074

Let $F_n$ be an $n$ by $n$ symmetric matrix whose entries are bounded by $n^{\gamma}$ for some $\gamma>0$. Consider a randomly perturbed matrix $M_n=F_n+X_n$, where $X_n$ is a random symmetric matrix whose upper diagonal entries $x_{ij}$ are iid copies of a random variable $\xi$. Under a very general assumption on $\xi$, we show that for any $B>0$ there exists $A>0$ such that $P(\sigma_n(M_n)\le n^{-A})\le n^{-B}$. The proof uses an inverse-type result concerning concentration of quadratic forms, which is of interest of its own., Comment: 36 p

Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that $M_n$ is non-singular with probability $1-O(n^{-C})$ for any positive constant $C$. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own., Comment: Some minor corrections in Section 10 of v1

We prove a version of Szemeredi's regularity lemma for subsets of a typical random set in F_p^n. As an application, a result on the distribution of three-term arithmetic progressions in sparse sets is discussed., Comment: 16 pages