arXiv subjects

Sean O'Rourke

Publications and source records attributed to Sean O'Rourke.

At least 37 records · Page 2Linked to original sources

arXiv2022

Spectrum of Heavy-Tailed Elliptic Random Matrices

An elliptic random matrix $X$ is a square matrix whose $(i,j)$-entry $X_{ij}$ is independent of the rest of the entries except possibly $X_{ji}$. Elliptic random matrices generalize Wigner matrices and non-Hermitian random matrices with independent entries. When the entries of an elliptic random matrix have mean zero and unit variance, the empirical spectral distribution is known to converge to the uniform distribution on the interior of an ellipse determined by the covariance of the mirrored entries. We consider elliptic random matrices whose entries fail to have two finite moments. Our main result shows that when the entries of an elliptic random matrix are in the domain of attraction of an $α$-stable random variable, for $0<α<2$, the empirical spectral measure converges, in probability, to a deterministic limit. This generalizes a result of Bordenave, Caputo, and Chafaï for heavy-tailed matrices with independent and identically distributed entries. The key elements of the proof are (i) a general bound on the least singular value of elliptic random matrices under no moment assumptions; and (ii) the convergence, in an appropriate sense, of the matrices to a random operator on the Poisson Weighted Infinite Tree.

math.PR↗

arXiv2021

Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth

We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists $τ\in (0,1)$ so that if the bandwidth of the matrix $X$ is at least $n^{1-τ}$ and the nonzero entries are iid random variables with mean zero and slightly more than four finite moments, then the limiting empirical eigenvalue distribution of $X$, when properly normalized, converges in probability to the uniform distribution on the unit disk in the complex plane. The key technical result is a least singular value bound for shifted random band block matrices with genuinely sublinear bandwidth, which improves on a result of Cook in the band matrix setting.

math.PR↗

arXiv2020

Sums of random polynomials with independent roots

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently from the probability measures $μ$ and $ν$ in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum $p+q$ as $n$ tends to infinity. The limiting distribution can be described by its logarithmic potential, which we show is the pointwise maximum of the logarithmic potentials of $μ$ and $ν$. More generally, we consider the sum of $m$ independent degree $n$ random polynomials when $m$ is fixed and $n$ tends to infinity. Our results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials.

math.PR↗

arXiv2020

A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation

Let $p_n:\mathbb{C} \rightarrow \mathbb{C}$ be a random complex polynomial whose roots are sampled i.i.d. from a radial distribution $u(r) r dr$ in the complex plane. A natural question is how the distribution of roots evolves under repeated (say $n/2-$times) differentiation of the polynomial. We conjecture a mean-field expansion for the evolution of $ψ(s) = u(s) s$ $$ \frac{\partial ψ}{\partial t} = \frac{\partial}{\partial x} \left( \left( \frac{1}{x} \int_{0}^{x} ψ(s) ds \right)^{-1} ψ(x) \right).$$ The evolution of $ψ(s) \equiv 1$ corresponds to the evolution of random Taylor polynomials $$ p_n(z) = \sum_{k=0}^{n}{ γ_k \frac{z^k}{k!}} \quad \mbox{where} \quad γ_k \sim \mathcal{N}_{\mathbb{C}}(0,1).$$ We discuss some numerical examples suggesting that this particular solution may be stable. We prove that the solution is linearly stable. The linear stability analysis reduces to the classical Hardy integral inequality. Many open problems are discussed.

math.AP↗

arXiv2020

Partial linear eigenvalue statistics for non-Hermitian random matrices

For an $n \times n$ independent-entry random matrix $X_n$ with eigenvalues $λ_1, \ldots, λ_n$, the seminal work of Rider and Silverstein asserts that the fluctuations of the linear eigenvalue statistics $\sum_{i=1}^n f(λ_i)$ converge to a Gaussian distribution for sufficiently nice test functions $f$. We study the fluctuations of $\sum_{i=1}^{n-K} f(λ_i)$, where $K$ randomly chosen eigenvalues have been removed from the sum. In this case, we identify the limiting distribution and show that it need not be Gaussian. Our results hold for the case when $K$ is fixed as well as the case when $K$ tends to infinity with $n$. The proof utilizes the predicted locations of the eigenvalues introduced by E. Meckes and M. Meckes. As a consequence of our methods, we obtain a rate of convergence for the empirical spectral distribution of $X_n$ to the circular law in Wasserstein distance, which may be of independent interest.

math.PR↗

arXiv2020

Eigenvectors and controllability of non-Hermitian random matrices and directed graphs

We study the eigenvectors and eigenvalues of random matrices with iid entries. Let $N$ be a random matrix with iid entries which have symmetric distribution. For each unit eigenvector $\mathbf{v}$ of $N$ our main results provide a small ball probability bound for linear combinations of the coordinates of $\mathbf{v}$. Our results generalize the works of Meehan and Nguyen as well as Touri and the second author for random symmetric matrices. Along the way, we provide an optimal estimate of the probability that an iid matrix has simple spectrum, improving a recent result of Ge. Our techniques also allow us to establish analogous results for the adjacency matrix of a random directed graph, and as an application we establish controllability properties of network control systems on directed graphs.

math.PR↗

arXiv2020

On the local pairing behavior of critical points and roots of random polynomials

We study the pairing between zeros and critical points of the polynomial $p_n(z) = \prod_{j=1}^n(z-X_j)$, whose roots $X_1, \ldots, X_n$ are complex-valued random variables. Under a regularity assumption, we show that if the roots are independent and identically distributed, the Wasserstein distance between the empirical distributions of roots and critical points of $p_n$ is on the order of $1/n$, up to logarithmic corrections. The proof relies on a careful construction of disjoint random Jordan curves in the complex plane, which allow us to naturally pair roots and nearby critical points. In addition, we establish asymptotic expansions to order $1/n^2$ for the locations of the nearest critical points to several fixed roots. This allows us to describe the joint limiting fluctuations of the critical points as $n$ tends to infinity, extending a recent result of Kabluchko and Seidel. Finally, we present a local law that describes the behavior of the critical points when the roots are neither independent nor identically distributed.

math.PR↗

arXiv2019

Random matrix products: Universality and least singular values

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 the centered linear spectral statistics of products of $M$ independent iid random matrices. This is done in two steps. First, we establish the result for product random matrices with Gaussian entries, and then extend to the general case of non-Gaussian entries by another moment matching argument. Prior to our result, Gaussian limits were known only for the case $M=1$. In a similar fashion, we establish Gaussian limits for the centered linear spectral statistics of products of independent truncated random unitary matrices. In both cases, we are able to obtain explicit expressions for the limiting variances. The main difficulty in our study is that the entries of the product matrix are no longer independent. Our key technical lemma is a lower bound on the least singular value of the translated linearization matrix associated with the product of $M$ normalized independent random matrices with independent and identically distributed subgaussian entries. This lemma is of independent interest.

math.PR↗

arXiv2019

Gaussian fluctuations for linear eigenvalue statistics of products of independent iid random matrices

Consider the product $X = X_{1}\cdots X_{m}$ of $m$ independent $n\times n$ iid random matrices. When $m$ is fixed and the dimension $n$ tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of $X$ for analytic test functions. We show that the limiting variance is universal in the sense that it does not depend on $m$ (the number of factor matrices) or on the distribution of the entries of the matrices. The main result generalizes and improves upon previous limit statements for the linear spectral statistics of a single iid matrix by Rider and Silverstein as well as Renfrew and the second author.

math.PR↗

arXiv2019

Eigenvector Delocalization for Non-Hermitian Random Matrices and Applications

Improving upon results of Rudelson and Vershynin, we establish delocalization bounds for eigenvectors of independent-entry random matrices. In particular, we show that with high probability every eigenvector is delocalized, meaning any subset of its coordinates carries an appropriate proportion of its mass. Our results hold for random matrices with genuinely complex as well as real entries. In both cases, our bounds match numerical simulations, up to lower order terms, indicating the optimality of our results. As an application of our methods, we also establish delocalization bounds for normal vectors to random hyperplanes. The proofs of our main results rely on a least singular value bound for genuinely complex rectangular random matrices, which generalizes a previous bound due to the first author, and may be of independent interest.

math.PR↗

arXiv2018

Low-degree factors of random polynomials

Motivated by the question of whether a random polynomial with integer coefficients is likely to be irreducible, we study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers, which is equivalent to having a low-degree algebraic root. It is known in certain cases that random polynomials with integer coefficients are very likely to be irreducible, and our project can be viewed as part of the general program of testing whether this is a universal behavior exhibited by many random polynomial models. Our main result shows that pointwise delocalization of the roots of a random polynomial can be used to imply that the polynomial is unlikely to have a low-degree factor over the integers. We apply our main result to a number of models of random polynomials, including characteristic polynomials of random matrices, where strong delocalization results are known. Studying a variety of random matrix models---including iid matrices, symmetric matrices, elliptical matrices, and adjacency matrices of random graphs and digraphs---we show that, for a random square matrix with integer entries, the characteristic polynomial is unlikely to have a low-degree factor over the integers, which is equivalent to the matrix having an eigenvalue that is algebraic with low degree. Having a low-degree algebraic eigenvalue generalizes the questions of whether the matrix has a rational eigenvalue and whether the matrix is singular (i.e., has an eigenvalue equal to zero).

math.PR↗

arXiv2018

Constrained Least Squares, SDP, and QCQP Perspectives on Joint Biconvex Radar Receiver and Waveform design

Joint radar receive filter and waveform design is non-convex, but is individually convex for a fixed receiver filter while optimizing the waveform, and vice versa. Such classes of problems are fre- quently encountered in optimization, and are referred to biconvex programs. Alternating minimization (AM) is perhaps the most popu- lar, effective, and simplest algorithm that can deal with bi-convexity. In this paper we consider new perspectives on this problem via older, well established problems in the optimization literature. It is shown here specifically that the radar waveform optimization may be cast as constrained least squares, semi-definite programs (SDP), and quadratically constrained quadratic programs (QCQP). The bi-convex constraint introduces sets which vary for each iteration in the alternat- ing minimization. We prove convergence of alternating minimization for biconvex problems with biconvex constraints by showing the equivalence of this to a biconvex problem with constrained Cartesian product convex sets but for convex hulls of small diameter.

eess.SP↗

arXiv2017

Outliers in the spectrum for products of independent random matrices

For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations.

math.PR↗

arXiv2017

Pairing between zeros and critical points of random polynomials with independent roots

Let $p_n$ be a random, degree $n$ polynomial whose roots are chosen independently according to the probability measure $μ$ on the complex plane. For a deterministic point $ξ$ lying outside the support of $μ$, we show that almost surely the polynomial $q_n(z):=p_n(z)(z - ξ)$ has a critical point at distance $O(1/n)$ from $ξ$. In other words, conditioning the random polynomials $p_n$ to have a root at $ξ$, almost surely forces a critical point near $ξ$. More generally, we prove an analogous result for the critical points of $q_n(z):=p_n(z)(z - ξ_1)\cdots (z - ξ_k)$, where $ξ_1, \ldots, ξ_k$ are deterministic. In addition, when $k=o(n)$, we show that the empirical distribution constructed from the critical points of $q_n$ converges to $μ$ in probability as the degree tends to infinity, extending a recent result of Kabluchko.

math.PR↗

arXiv2016

On a conjecture of Godsil concerning controllable random graphs

It is conjectured by Godsil that the relative number of controllable graphs compared to the total number of simple graphs on n vertices approaches one as n tends to infinity. We prove that this conjecture is true. More generally, our methods show that the linear system formed from the pair (W, b) is controllable for a large class of Wigner random matrices W and deterministic vectors b. The proof relies on recent advances in Littlewood-Offord theory developed by Rudelson and Vershynin.

math.OC↗

arXiv2016

Eigenvectors of random matrices: A survey

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.

math.PR↗

arXiv2016

Spectra of nearly Hermitian random matrices

We consider the eigenvalues and eigenvectors of matrices of the form M + P, where M is an n by n Wigner random matrix and P is an arbitrary n by n deterministic matrix with low rank. In general, we show that none of the eigenvalues of M + P need be real, even when P has rank one. We also show that, except for a few outlier eigenvalues, most of the eigenvalues of M + P are within 1/n of the real line, up to small order corrections. We also prove a new result quantifying the outlier eigenvalues for multiplicative perturbations of the form S ( I + P ), where S is a sample covariance matrix and I is the identity matrix. We extend our result showing all eigenvalues except the outliers are close to the real line to this case as well. As an application, we study the critical points of the characteristic polynomials of nearly Hermitian random matrices.

math.PR↗

arXiv2015

Critical points of random polynomials and characteristic polynomials of random matrices

Let $p_n$ be the characteristic polynomial of an $n \times n$ random matrix drawn from one of the compact classical matrix groups. We show that the critical points of $p_n$ converge to the uniform distribution on the unit circle as $n$ tends to infinity. More generally, we show the same limit for a class of random polynomials whose roots lie on the unit circle. Our results extend the work of Pemantle-Rivin and Kabluchko to the setting where the roots are neither independent nor identically distributed.

math.PR↗