arXiv subjects

Nicholas A. Cook

Publications and source records attributed to Nicholas A. Cook.

18 recordsLinked to original sources

arXiv2025

Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$

We prove that the empirical law of eigenvalues of Brownian motion on the Lie Group $\mathrm{GL}(N,\mathbb{C})$ converges almost surely to a deterministic probability measure, characterized by a free stochastic differential equation. This fully resolves a conjecture made by Philippe Biane in 1997. Our analysis includes a family $\{B=B_{\rho,\zeta}\colon |\zeta|<\rho\}$ of nondegenerate diffusion processes on $\mathrm{GL}(N,\mathbb{C})$ whose laws are invariant under unitary conjugation, with initial distributions assumed to be uniformly bounded and invertible. The crux of our analysis is a strong quantitative approximation of Brownian motion $B(t)$ on $\mathrm{GL}(N,\mathbb{C})$ for small $t$ by a single increment $I+W(t)$, where $W=W_{\rho,\zeta}$ is an elliptic Brownian motion in the Lie algebra $\mathfrak{gl}(N,\mathbb{C}) = \mathbb{M}_N(\mathbb{C})$. Specifically, for any $t\in[0,1]$ and $\delta>0$, \[ \mathbb{P}\left(\|B(t)-I-W(t)\|\geq \delta\right)\leq \left(C t/\delta\right)^{N^{2/3}} \] for a constant $C=C_\rho$. Leveraging independence of multiplicative increments of the Brownian motion then allows us to use powerful (anti-)concentration tools for Gaussian matrices to complete the Hermitization procedure for convergence of eigenvalues.

math.PR↗

arXiv2025

Upper tails for homomorphism counts in sparse random hypergraphs

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 hypergraph $\boldsymbol{G}$ exceeds its expectation by a constant factor. The problem was popularized by Janson and Ruci\'nski and, particularly in the case of graphs ($r=2$), has been a driving example in the development of nonlinear large deviations theory. Recent work of the first author with Dembo and Pham has accomplished the \emph{naive mean-field reduction step}, reducing the upper tail problem to an entropic variational problem on a space of weighted graphs. The latter was resolved for counts of $r$-uniform cliques and a certain linear 3-uniform hypergraph by Liu and Zhao, who also conjectured a general formula. We confirm their conjecture for other classes of hypergraphs, including complete $r$-partite $r$-graphs, tight cycles, and the Fano plane. We also prove a general large deviation upper bound for counts of $r$-graphs $H$ satisfying certain edge covering properties.

math.CO↗

arXiv2023

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

We establish precise upper-tail asymptotics and large deviation principles for the rightmost eigenvalue $\lambda_1$ of Wigner matrices with sub-Gaussian entries. In contrast to the case of heavier tails, where deviations of $\lambda_1$ are due to the appearance of a few large entries, and the sharp sub-Gaussian case that is governed by the collective deviation of entries in a delocalized rank-one pattern, we show that the general sub-Gaussian case is determined by a mixture of localized and delocalized effects. Our key result is a finite-$N$ approximation for the upper tail of $\lambda_1$ by an optimization problem involving \emph{restricted annealed free energies} for a spherical spin glass model. This new type of argument allows us to derive full large deviation principles when the log-Laplace transform of the entries' distribution $\mu$ has bounded second derivative, whereas previous results required much more restrictive assumptions, namely sharp sub-Gaussianity and symmetry, or only covered certain ranges of deviations. We show that the sharp sub-Gaussian condition characterizes measures $\mu$ for which the rate function coincides with that of the Gaussian Orthogonal Ensemble (GOE). When $\mu$ is not sharp sub-Gaussian, at a certain distance from the bulk of the spectrum there is a transition from the GOE rate function to a non-universal rate function depending on $\mu$, and this transition coincides with the onset of a localization phenomenon for the associated eigenvector.

math.PR↗

arXiv2022

Typical structure of sparse exponential random graph models

We consider general Exponential Random Graph Models (ERGMs) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs $F_k$. Whereas previous work has shown a degeneracy phenomenon in dense ERGMs, we show this can be cured by raising the sufficient statistics to a fractional power. We rigorously establish the na\"ive mean-field approximation for the partition function of the corresponding Gibbs measures, and in case of "ferromagnetic" models with vanishing edge density show that typical samples resemble a typical Erd\H{o}s--R\'enyi graph with a planted clique and/or a planted complete bipartite graph of appropriate sizes. We establish such behavior also for the conditional structure of the Erd\H{o}s--R\'enyi graph in the large deviations regime for excess $F_k$-homomorphism counts. These structural results are obtained by combining quantitative large deviation principles, established in previous works, with a novel stability form of a result of [5] on the asymptotic solution for the associated entropic variational problem. A technical ingredient of independent interest is a stability form of Finner's generalized H\"older inequality.

math.PR↗

arXiv2021

Universality of Poisson limits for moduli of roots of Kac polynomials

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.

math.PR↗

arXiv2021

Regularity method and large deviation principles for the Erd\H{o}s--R\'enyi hypergraph

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 of homomorphism counts in the $r$-uniform Erd\H{o}s--R\'enyi hypergraph for any fixed $r\ge 2$, generalizing and improving on previous results for the Erd\H{o}s--R\'enyi graph ($r=2$). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields tail asymptotics for other nonlinear functionals, such as induced homomorphism counts.

math.PR↗

arXiv2021

Universality of the minimum modulus for random trigonometric polynomials

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.

math.PR↗

arXiv2020

Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices

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 $P^N =\mathfrak{p}(X_1^N,\dots, X_n^N)$ to the Brown measure of $\mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,\dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.

math.PR↗

arXiv2020

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

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. In the companion article Cook et al., we considered the empirical spectral distribution $\mu_n^Y$ of the rescaled entry-wise product \[ Y_n = \frac 1{\sqrt{n}} A_n\odot X_n = \left(\frac1{\sqrt{n}} \sigma_{ij}X_{ij}\right) \] and provided a deterministic sequence of probability measures $\mu_n$ such that the difference $\mu^Y_n - \mu_n$ converges weakly in probability to the zero measure. A key feature in Cook et al. was to allow some of the entries $\sigma_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. In the present article, we provide more information on the sequence $(\mu_n)$, described by a family of Master Equations. We consider these equations in important special cases such as separable variance profiles $\sigma^2_{ij}=d_i \widetilde d_j$ and sampled variance profiles $\sigma^2_{ij} = \sigma^2\left(\frac in, \frac jn \right)$ where $(x,y)\mapsto \sigma^2(x,y)$ is a given function on $[0,1]^2$. Associate examples are provided where $\mu_n^Y$ converges to a genuine limit. We study $\mu_n$'s behavior at zero and provide examples where $\mu_n$'s density is bounded, blows up, or vanishes while an atom appears. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al., we prove that except maybe in zero, $\mu_n$ admits a positive density on the centered disc of radius $\sqrt{\rho(V_n)}$, where $V_n=(\frac 1n \sigma_{ij}^2)$ and $\rho(V_n)$ is its spectral radius.

math.PR↗

arXiv2018

Large deviations of subgraph counts for sparse Erd\H{o}s--R\'enyi graphs

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 Erd\H{o}s--R\'enyi graph $G(n,p)$ exceeds its expectation by a factor $1+u$, assuming $n^{-\kappa(H)}\ll p\ll1$, with $\kappa(H) = 1/(2\Delta)$, where $\Delta\ge 1$ is the maximum degree of $H$. This improves on a previous result of Chatterjee and the second author, who obtained $\kappa(H)=c/(\Delta|E|)$ for a constant $c>0$. Moreover, for the case of cycle counts we can take $\kappa$ as large as $1/2$. We additionally obtain the sharp upper tail for Schatten norms of the adjacency matrix, as well as the sharp lower tail for counts of graphs for which Sidorenko's conjecture holds. As a key step, we establish quantitative versions of Szemer\'edi's regularity lemma and the counting lemma, suitable for the analysis of random graphs in the large deviations regime.

math.PR↗

arXiv2018

Lower bounds for the smallest singular value of structured random matrices

We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[ M = A\circ X + B = (a_{ij}ξ_{ij} + b_{ij}) \] where $X=(ξ_{ij})$ has iid centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B= Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As a byproduct of our methods we can also handle general perturbations $B$ under additional hypotheses on $A$, which translate to connectivity hypotheses on an associated graph. In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses. Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular Szemerédi's Regularity Lemma, and a version of the Restricted Invertibility Theorem due to Spielman and Srivastava.

math.PR↗

arXiv2017

The circular law for random regular digraphs with random edge weights

We consider random $n\times n$ matrices of the form $Y_n=\frac1{\sqrt{d}}A_n\circ X_n$, where $A_n$ is the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices, with $d=\lfloor p n\rfloor$ for some fixed $p \in (0,1)$, and $X_n$ is an $n\times n$ matrix of iid centered random variables with unit variance and finite $4+η$-th moment (here $\circ$ denotes the matrix Hadamard product). We show that as $n\to \infty$, the empirical spectral distribution of $Y_n$ converges weakly in probability to the normalized Lebesgue measure on the unit disk.

math.PR↗

arXiv2017

The circular law for random regular digraphs

Let $\log^Cn\le d\le n/2$ 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 that as $n$ tends to infinity, the empirical spectral distribution of $A_n$, suitably rescaled, is governed by the Circular Law. A key step is to obtain quantitative lower tail bounds for the smallest singular value of additive perturbations of $A_n$.

math.PR↗

arXiv2017

Size biased couplings and the spectral gap for random regular graphs

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 $d$ is fixed independent of $n$, then $λ=2\sqrt{d-1} +o(1)$ with high probability. In the present work we show that $λ=O(\sqrt{d})$ continues to hold with high probability as long as $d=O(n^{2/3})$, making progress towards a conjecture of Vu that the bound holds for all $1\le d\le n/2$. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at $d=o(n^{1/2})$. We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on $d$-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.

math.PR↗

arXiv2016

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

math.PR↗

arXiv2015

On the singularity of adjacency matrices for random regular digraphs

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 $\min(d,n-d)\ge C\log^2n$ for a sufficiently large constant $C>0$. The proof makes use of a coupling of random regular digraphs formed by "shuffling" the neighborhood of a pair of vertices, as well as concentration results for the distribution of edges recently obtained by the author (arXiv:1410.5595). We also apply our general approach to prove a.a.s.\ invertibility of Hadamard products $Σ\circ Ξ$, where $Ξ$ is a matrix of iid uniform $\pm1$ signs, and $Σ$ is a 0/1 matrix whose associated digraph satisfies certain "expansion" properties.

math.PR↗

arXiv2015

Discrepancy properties for random regular digraphs

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 edges essentially matching results for Erdős-Rényi digraphs obtained from Chernoff-type bounds. The proofs make use of the method of exchangeable pairs, developed for concentration of measure by Chatterjee. Exchangeable pairs are constructed using two involutions on the set of regular digraphs: a well-known "simple switching" operation, as well as a novel "reflection" operation.

math.CO↗

arXiv2015

Dense random regular digraphs: singularity of the adjacency matrix

Fix $c\in (0,1)$ and let $Γ$ 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 $Γ$ is invertible with high probability. The proof uses a couplings approach based on the switchings method of McKay and Wormald. We also rely on discrepancy properties for the distribution of edges in $Γ$, recently proved by the author, to overcome certain difficulties stemming from the dependencies between the entries of $M$.

math.PR↗