Nicholas A. Cook

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 Ruciński 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.

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.

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.

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.

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 $\mu_n^Y$ of the rescaled entry-wise product \[ Y_n = \left(\frac1{\sqrt{n}} \sigma_{ij}X_{ij}\right). \] For our main result we provide a deterministic sequence of probability measures $\mu_n$, each described by a family of Master Equations, such that the difference $\mu^Y_n - \mu_n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries $\sigma_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. An important step is to obtain quantitative bounds on the solutions to an associate system of Schwinger--Dyson equations, which we accomplish in the general sparse setting using a novel graphical bootstrap argument.

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.

Let $\log^{C}n\le d\le n/2$ 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 $A_{n}$. Soit $\log^{C}n\le d\le n/2$ pour une constante suffisamment grande $C>0$. Notons $A_{n}$ la matrice d’adjacence d’un graphe dirigé aléatoire $d$-régulier sur $n$ sommets. Nous montrons que lorsque $n$ tend vers l’infini, la distribution empirique des valeurs propres de $A_{n}$, convenablement normalisée, suit la loi du cercle. Une étape cruciale consiste à obtenir une borne inférieure quantitative asymptotique pour la plus petite valeur singulière de perturbations additives de $A_{n}$.