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Sean O'Rourke

Publications and source records attributed to Sean O'Rourke.

50 records · Page 3Linked to original sources
arXiv2015

Controllability of random systems: Universality and minimal controllability

For a large class of random matrices $A$ and vectors $b$, we show that linear systems formed from the pair $(A,b)$ are controllable with high probability. Despite the fact that minimal controllability problems are, in general, NP-hard, we establish universality results for the minimal controllability of random systems. Our proof relies on the recent developments of Nguyen-Tao-Vu concerning gaps between eigenvalues of random matrices.

math.PR↗
arXiv2015

Central limit theorem for linear eigenvalue statistics of elliptic random matrices

We consider a class of elliptic random matrices which generalize two classical ensembles from random matrix theory: Wigner matrices and random matrices with iid entries. In particular, we establish a central limit theorem for linear eigenvalue statistics of real elliptic random matrices under the assumption that the test functions are analytic. As a corollary, we extend the results of Rider and Silverstein to real iid random matrices.

math.PR↗
arXiv2015

Products of independent elliptic random matrices

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. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. Our result also generalizes earlier results of Götze-Tikhomirov and O'Rourke-Soshnikov concerning the product of independent iid random matrices.

math.PR↗
arXiv2014

The Elliptic Law

We show that, under some general assumptions on the entries of a random complex $n \times n$ matrix $X_n$, the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges to the uniform law of an ellipsoid as $n$ tends to infinity. This generalizes the well-known circular law in random matrix theory.

math.PR↗
arXiv2014

On the concentration of random multilinear forms and the universality of random block matrices

The circular law asserts that if $X_n$ is a $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges almost surely to the uniform distribution on the unit disk as $n$ tends to infinity. Answering a question of Tao, we prove the circular law for a general class of random block matrices with dependent entries. The proof relies on an inverse-type result for the concentration of linear operators and multilinear forms.

math.PR↗
arXiv2014

Low rank perturbations of large elliptic random matrices

We study the asymptotic behavior of outliers in the spectrum of bounded rank perturbations of large random matrices. In particular, we consider perturbations of elliptic random matrices which generalize both Wigner random matrices and non-Hermitian random matrices with iid entries. As a consequence, we recover the results of Capitaine, Donati-Martin, and Féral for perturbed Wigner matrices as well as the results of Tao for perturbed random matrices with iid entries. Along the way, we prove a number of interesting results concerning elliptic random matrices whose entries have finite fourth moment; these results include a bound on the least singular value and the asymptotic behavior of the spectral radius.

math.PR↗
arXiv2014

Universality of local eigenvalue statistics in random matrices with external source

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 matrices $D_n$. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones. The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.

math.PR↗
arXiv2013

Partial Linear Eigenvalue Statistics for Wigner and Sample Covariance Random Matrices

Let $M_n$ be a $n \times n$ Wigner or sample covariance random matrix, and let $μ_1(M_n), μ_2(M_n), ..., μ_n(M_n)$ denote the unordered eigenvalues of $M_n$. We study the fluctuations of the partial linear eigenvalue statistics $$ \sum_{i=1}^{n-k} f(μ_i(M_n)) $$ as $n \rightarrow \infty$ for sufficiently nice test functions $f$. We consider both the case when $k$ is fixed and when $\min{k,n-k}$ tends to infinity with $n$.

math.PR↗
arXiv2012

Fluctuations of Matrix Entries of Analytic Functions of Non-Hermitian Random Matrices

Consider an $n \times n$ non-Hermitian random matrix $M_n$ whose entries are independent real random variables. Under suitable conditions on the entries, we study the fluctuations of the entries of $f(M_n)$ as $n$ tends to infinity, where $f$ is analytic on an appropriate domain. This extends the results for symmetric random matrices to the non-Hermitian case.

math.PR↗
arXiv2011

On Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices with Non-Identically Distributed Entries

In this note, we extend the results about the fluctuations of the matrix entries of regular functions of Wigner random matrices obtained in arXiv:1103.3731 [math.PR] to Wigner matrices with non-i.i.d. entries provided certain Lindeberg type conditions for the fourth moments of the off-diagonal entries and the second moments of the diagonal entries are satisfied. In addition, we relax our conditions on the test functions and require that for some $s>3 \ \int (1+|k|)^{2s}\*|\hat{f}(k)|^2 \* dk <\infty.$

math.PR↗
arXiv2011

Products of Independent Non-Hermitian Random Matrices

For fixed $m>1$, we consider $m$ independent $n \times n$ non-Hermitian random matrices $X_1, ..., X_m$ with i.i.d. centered entries with a finite $(2+η)$-th moment, $ η>0.$ As $n$ tends to infinity, we show that the empirical spectral distribution of $n^{-m/2} \*X_1 X_2 ... X_m$ converges, with probability 1, to a non-random, rotationally invariant distribution with compact support in the complex plane. The limiting distribution is the $m$-th power of the circular law.

math.PR↗
arXiv2010

Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an $n \times n$ matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let $x_k$ denote eigenvalue number $k$. Under the condition that both $k$ and $n-k$ tend to infinity with $n$, we show that $x_k$ is normally distributed in the limit. We also consider the joint limit distribution of $m$ eigenvalues from the GOE or GSE with similar conditions on the indices. The result is an $m$-dimensional normal distribution. Using a recent universality result by Tao and Vu, we extend our results to a class of Wigner real symmetric matrices with non-Gaussian entries that have an exponentially decaying distribution and whose first four moments match the Gaussian moments.

math.PR↗