Journal ArticleAnnals of Applied Probability · June 1, 2024
We consider general exponential random graph models (ERGMs) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs Fk. Whereas previous work has shown a degeneracy phenomenon in dense ERGMs, we show thi ...
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Journal ArticleDuke 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 ...
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Journal ArticleInternational 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 ...
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Journal ArticleJournal 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 ...
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Journal ArticleAnnales 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 ...
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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 ...
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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 ...
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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 ...
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Journal ArticleDiscrete 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 ...
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Journal ArticleAdvances 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 ...
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Journal ArticleCommunications 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 ...
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Journal ArticleAnnales 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 ...
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Journal ArticleAnnals 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 ...
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Journal ArticleElectronic 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 ...
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Journal ArticleElectronic 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 ...
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Journal ArticleRandom 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 ...
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Journal ArticleProbability 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 ...
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Journal ArticleRandom 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 ...
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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_ ...
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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 ...
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Journal ArticleTo 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,\ ...
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