Nicholas A. Cook

activity

20142025

most citedSize biased couplings and the spectral gap for random regular graphs

42 citations · 103 across the 13 of their papers we have counts for

collaborators

19 papers

math.CO2025

Upper tails for homomorphism counts in sparse random hypergraphs

Nicholas A. Cook, Nguyen Nguyen

The "infamous upper tail problem" for $r$ -uniform hypergraphs is to estimate the probability that the number of copies of a fixed hypergraph $H$ in a large binomial $r$ -uniform hyp…

math.PR2023★ 3 cited

Full large deviation principles for the largest eigenvalue of sub-Gaussian Wigner matrices

Nicholas A. Cook, Raphael Ducatez, Alice Guionnet

We establish precise upper-tail asymptotics and large deviation principles for the rightmost eigenvalue $λ_{1}$ of Wigner matrices with sub-Gaussian entries. In contrast to the case…

math.PR2022

Typical structure of sparse exponential random graph models

Nicholas A. Cook, Amir Dembo

We consider general Exponential Random Graph Models (ERGMs) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs $F_{k}$ . Wher…

math.PR2021

Universality of Poisson limits for moduli of roots of Kac polynomials

Nicholas A. Cook, Hoi H. Nguyen, Oren Yakir +1

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 $n$ , c…

math.PR2021★ 1 cited

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

Nicholas A. Cook, Amir Dembo, Huy Tuan Pham

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 mai…

math.PR2021★ 2 cited

Universality of the minimum modulus for random trigonometric polynomials

Nicholas A. Cook, Hoi H. Nguyen

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…

math.PR2020

Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices

Nicholas A. Cook, Alice Guionnet, Jonathan Husson

For a fixed quadratic polynomial $p$ in $n$ non-commuting variables, and $n$ independent $N \times N$ complex Ginibre matrices $X_{1}^{N}, \dots, X_{n}^{N}$ , we establish the conv…

math.PR2020★ 5 cited

Non-Hermitian random matrices with a variance profile (II): properties and examples

Nicholas A. Cook, Walid Hachem, Jamal Najim +1

For each $n$ , let $A_{n} = (σ_{ij})$ 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. In the…

math.PR2018★ 1 cited

Large deviations of subgraph counts for sparse Erdős--Rényi graphs

Nicholas A. Cook, Amir Dembo

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 th…

math.PR2018

Maximum of the characteristic polynomial for a random permutation matrix

Nicholas Cook, Ofer Zeitouni

Let $P_{N}$ be a uniform random $N \times N$ permutation matrix and let $χ_{N} (z) = det (z I_{N} - P_{N})$ denote its characteristic polynomial. We prove a law of large numbers for the maximum…

math.PR2017

Circular law for the sum of random permutation matrices

Anirban Basak, Nicholas Cook, Ofer Zeitouni

Let $P_{n}^{1}, \dots, P_{n}^{d}$ be $n \times n$ permutation matrices drawn independently and uniformly at random, and set $S_{n}^{d} := \sum_{ℓ = 1}^{d} P_{n}^{ℓ}$ . We show that if $\log^{12}n/(\l…

math.PR2017

The circular law for random regular digraphs

Nicholas A. Cook

Let $lo g^{C} n \leq d \leq n /2$ for a sufficiently large constant $C > 0$ and let $A_{n}$ denote the adjacency matrix of a uniform random $d$ -regular directed graph on $n$ vertices. We prove…

math.PR2016★ 34 cited

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

Nicholas A. Cook, Walid Hachem, Jamal Najim +1

For each $n$ , let $A_{n} = (σ_{ij})$ 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 stu…

math.PR2016

Lower bounds for the smallest singular value of structured random matrices

Nicholas A. Cook

We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n \times n$ ma…

math.PR2015★ 42 cited

Size biased couplings and the spectral gap for random regular graphs

Nicholas A. Cook, Larry Goldstein, Tobias Johnson

Let $λ$ be the second largest eigenvalue in absolute value of a uniform random $d$ -regular graph on $n$ vertices. It was famously conjectured by Alon and proved by Friedman that if…

math.PR2015★ 8 cited

The circular law for random regular digraphs with random edge weights

Nicholas A. Cook

We consider random $n \times n$ matrices of the form $Y_{n} = \frac{1}{d} A_{n} \circ X_{n}$ , where $A_{n}$ is the adjacency matrix of a uniform random $d$ -regular directed graph on $n$ ve…

math.PR2014★ 5 cited

On the singularity of adjacency matrices for random regular digraphs

Nicholas A. Cook

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)\…

math.CO2014

Discrepancy properties for random regular digraphs

Nicholas A. Cook

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 co…

math.PR2014★ 2 cited

Dense random regular digraphs: singularity of the adjacency matrix

Nicholas A. Cook

Fix $c \in (0, 1)$ and let $Γ$ be a $⌊ c n ⌋$ -regular digraph on $n$ vertices drawn uniformly at random. We prove that when $n$ is large, the (non-symmetric) adjacency mat…