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Nicholas A Cook

Assistant Professor of Mathematics
Mathematics
120 Science Drive, Durham, NC 27708

Selected Publications


REGULARITY METHOD AND LARGE DEVIATION PRINCIPLES FOR THE ERDŐS–RÉNYI HYPERGRAPH

Journal Article Duke Mathematical Journal · April 1, 2024 We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails o ... Full text Cite

Universality of Poisson Limits for Moduli of Roots of Kac Polynomials

Journal Article International Mathematics Research Notices · April 1, 2023 We give a new proof of a recent resolution [18] by Michelen and Sahasrabudhe of a conjecture of Shepp and Vanderbei [19] 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 ... Full text Cite

Non-Hermitian Random Matrices with a Variance Profile (II): Properties and Examples

Journal Article Journal of Theoretical Probability · December 1, 2022 For each n, let An= (σij) be an n× n deterministic matrix and let Xn= (Xij) be an n× n random matrix with i.i.d. centered entries of unit variance. In the companion article (Cook et al. in Electron J Probab 23:Paper No. 110, 61, 2018), we considered the em ... Full text Cite

Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices

Journal Article Annales de l'institut Henri Poincare (B) Probability and Statistics · November 1, 2022 For a fixed quadratic polynomial p in n non-commuting variables, and n independent N × N complex Ginibre matrices XN1, ⋯, XNn, we establish the convergence of the empirical measure of the eigenvalues of PN = p(XN1, ⋯, XNn) to the Brown measure of p evaluat ... Full text Cite

Universality of Poisson limits for moduli of roots of Kac polynomials

Journal Article · May 18, 2021 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. W ... Link to item Cite

Regularity method and large deviation principles for the Erdős--Rényi hypergraph

Journal Article · February 17, 2021 We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails o ... Link to item Cite

Universality of the minimum modulus for random trigonometric polynomials

Journal Article · January 18, 2021 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 dis ... Link to item Cite

Universality of the minimum modulus for random trigonometric polynomials

Journal Article Discrete Analysis · January 1, 2021 It has been shown in [YZ] 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 o ... Full text Cite

Large deviations of subgraph counts for sparse Erdős–Rényi graphs

Journal Article Advances in Mathematics · October 28, 2020 For any fixed simple graph H=(V,E) and any fixed u>0, we establish the leading order of the exponential rate function for the probability that the number of copies of H in the Erdős–Rényi graph G(n,p) exceeds its expectation by a factor 1+u, assuming n−κ(H ... Full text Cite

Maximum of the Characteristic Polynomial for a Random Permutation Matrix

Journal Article Communications on Pure and Applied Mathematics · August 1, 2020 Let PN be a uniform random N × N permutation matrix and let χN(z) = det(zIN − PN) denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of χN on the unit circle, specifically, (Formula presented.) with probability te ... Full text Cite

The Circular Law for random regular digraphs

Journal Article Annales de l'institut Henri Poincare (B) Probability and Statistics · January 1, 2019 Let logC n ≤ d ≤ n/2 for a sufficiently large constant C > 0 and let An denote the adjacency matrix of a uniform random d-regular directed graph on n vertices. We prove that as n tends to infinity, the empirical spectral distribution of An, suitably rescal ... Full text Cite

Lower bounds for the smallest singular value of structured random matrices

Journal Article Annals of Probability · November 1, 2018 We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider n× n matrices with complex entries of the form M = A ⇆ X + B = (aij ξij +bij), where X = (ξ ... Full text Cite

Non-hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs

Journal Article Electronic Journal of Probability · January 1, 2018 For each n, let An = (σij) be an n × n deterministic matrix and let Xn = (Xij) be an n × n random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution µYn of the rescaled entry-wise p ... Full text Cite

Circular law for the sum of random permutation matrices

Journal Article Electronic Journal of Probability · January 1, 2018 Let Pn1, …, Pnd be n × n permutation matrices drawn independently and uniformly at random, and set Snd := ∑ℓ-1d Pnℓ. We show that if log12n/(log log n)4 ≤ d = O(n), then the empirical spectral distribution of Snd/√d converges weakly to the circular law in ... Full text Cite

Size biased couplings and the spectral gap for random regular graphs

Journal Article The Annals of Probability · January 1, 2018 Full text Cite

The circular law for random regular digraphs with random edge weights

Journal Article Random Matrices: Theory and Application · July 1, 2017 We consider random n × n matrices of the form Yn = 1 dAn Xn, where An is the adjacency matrix of a uniform random d-regular directed graph on n vertices, with d = ?pn? for some fixed p (0, 1), and Xn is an n × n matrix of i.i.d. centered random variables w ... Full text Cite

On the singularity of adjacency matrices for random regular digraphs

Journal Article Probability Theory and Related Fields · February 1, 2017 We prove that the (non-symmetric) adjacency matrix of a uniform random d-regular directed graph on n vertices is asymptotically almost surely invertible, assuming min (d, n- d) ≥ Clog 2n for a sufficiently large constant C> 0. The proof makes use of a coup ... Full text Cite

Discrepancy properties for random regular digraphs

Journal Article Random Structures and Algorithms · January 1, 2017 For the uniform random regular directed graph we prove concentration inequalities for (1) codegrees and (2) the number of edges passing from one set of vertices to another. As a consequence, we can deduce discrepancy properties for the distribution of edge ... Full text Cite

REGULARITY METHOD AND LARGE DEVIATION PRINCIPLES FOR THE ERDŐS–RÉNYI HYPERGRAPH

Journal Article Duke Mathematical Journal · April 1, 2024 We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails o ... Full text Cite

Universality of Poisson Limits for Moduli of Roots of Kac Polynomials

Journal Article International Mathematics Research Notices · April 1, 2023 We give a new proof of a recent resolution [18] by Michelen and Sahasrabudhe of a conjecture of Shepp and Vanderbei [19] 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 ... Full text Cite

Non-Hermitian Random Matrices with a Variance Profile (II): Properties and Examples

Journal Article Journal of Theoretical Probability · December 1, 2022 For each n, let An= (σij) be an n× n deterministic matrix and let Xn= (Xij) be an n× n random matrix with i.i.d. centered entries of unit variance. In the companion article (Cook et al. in Electron J Probab 23:Paper No. 110, 61, 2018), we considered the em ... Full text Cite

Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices

Journal Article Annales de l'institut Henri Poincare (B) Probability and Statistics · November 1, 2022 For a fixed quadratic polynomial p in n non-commuting variables, and n independent N × N complex Ginibre matrices XN1, ⋯, XNn, we establish the convergence of the empirical measure of the eigenvalues of PN = p(XN1, ⋯, XNn) to the Brown measure of p evaluat ... Full text Cite

Universality of Poisson limits for moduli of roots of Kac polynomials

Journal Article · May 18, 2021 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. W ... Link to item Cite

Regularity method and large deviation principles for the Erdős--Rényi hypergraph

Journal Article · February 17, 2021 We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails o ... Link to item Cite

Universality of the minimum modulus for random trigonometric polynomials

Journal Article · January 18, 2021 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 dis ... Link to item Cite

Universality of the minimum modulus for random trigonometric polynomials

Journal Article Discrete Analysis · January 1, 2021 It has been shown in [YZ] 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 o ... Full text Cite

Large deviations of subgraph counts for sparse Erdős–Rényi graphs

Journal Article Advances in Mathematics · October 28, 2020 For any fixed simple graph H=(V,E) and any fixed u>0, we establish the leading order of the exponential rate function for the probability that the number of copies of H in the Erdős–Rényi graph G(n,p) exceeds its expectation by a factor 1+u, assuming n−κ(H ... Full text Cite

Maximum of the Characteristic Polynomial for a Random Permutation Matrix

Journal Article Communications on Pure and Applied Mathematics · August 1, 2020 Let PN be a uniform random N × N permutation matrix and let χN(z) = det(zIN − PN) denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of χN on the unit circle, specifically, (Formula presented.) with probability te ... Full text Cite

The Circular Law for random regular digraphs

Journal Article Annales de l'institut Henri Poincare (B) Probability and Statistics · January 1, 2019 Let logC n ≤ d ≤ n/2 for a sufficiently large constant C > 0 and let An denote the adjacency matrix of a uniform random d-regular directed graph on n vertices. We prove that as n tends to infinity, the empirical spectral distribution of An, suitably rescal ... Full text Cite

Lower bounds for the smallest singular value of structured random matrices

Journal Article Annals of Probability · November 1, 2018 We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider n× n matrices with complex entries of the form M = A ⇆ X + B = (aij ξij +bij), where X = (ξ ... Full text Cite

Non-hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs

Journal Article Electronic Journal of Probability · January 1, 2018 For each n, let An = (σij) be an n × n deterministic matrix and let Xn = (Xij) be an n × n random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution µYn of the rescaled entry-wise p ... Full text Cite

Circular law for the sum of random permutation matrices

Journal Article Electronic Journal of Probability · January 1, 2018 Let Pn1, …, Pnd be n × n permutation matrices drawn independently and uniformly at random, and set Snd := ∑ℓ-1d Pnℓ. We show that if log12n/(log log n)4 ≤ d = O(n), then the empirical spectral distribution of Snd/√d converges weakly to the circular law in ... Full text Cite

Size biased couplings and the spectral gap for random regular graphs

Journal Article The Annals of Probability · January 1, 2018 Full text Cite

The circular law for random regular digraphs with random edge weights

Journal Article Random Matrices: Theory and Application · July 1, 2017 We consider random n × n matrices of the form Yn = 1 dAn Xn, where An is the adjacency matrix of a uniform random d-regular directed graph on n vertices, with d = ?pn? for some fixed p (0, 1), and Xn is an n × n matrix of i.i.d. centered random variables w ... Full text Cite

On the singularity of adjacency matrices for random regular digraphs

Journal Article Probability Theory and Related Fields · February 1, 2017 We prove that the (non-symmetric) adjacency matrix of a uniform random d-regular directed graph on n vertices is asymptotically almost surely invertible, assuming min (d, n- d) ≥ Clog 2n for a sufficiently large constant C> 0. The proof makes use of a coup ... Full text Cite

Discrepancy properties for random regular digraphs

Journal Article Random Structures and Algorithms · January 1, 2017 For the uniform random regular directed graph we prove concentration inequalities for (1) codegrees and (2) the number of edges passing from one set of vertices to another. As a consequence, we can deduce discrepancy properties for the distribution of edge ... Full text Cite

Non-Hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs

Journal Article · December 13, 2016 For each $n$, let $A_n=(\sigma_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral distribution $\mu_ ... Link to item Cite

Dense random regular digraphs: singularity of the adjacency matrix

Journal Article · March 24, 2014 Fix $c\in (0,1)$ and let $\Gamma$ be a $\lfloor c n\rfloor$-regular digraph on $n$ vertices drawn uniformly at random. We prove that when $n$ is large, the (non-symmetric) adjacency matrix $M$ of $\Gamma$ is invertible with high probability. The proof uses ... Link to item Cite

Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices

Journal Article To appear in Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques For a fixed quadratic polynomial $\mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =\mathfrak{p}(X_1^N,\ ... Link to item Cite