Let $d>1$ be fixed and let $A_n$ be an $n\times n$ matrix with independent $\Ber(d/n)$ entries. For every $0<r<\sqrt d$, we prove that, with high probability, a positive proportion of the eigenvalues of $A_n$ have modulus larger than $r$. Together with the known upper bound, this implies that the modulus of the second largest eigenvalue converges in probability to $\sqrt d$. Our proof works with the Brown measure $\mu_d$ of the adjacency operator of the directed Poisson--Galton--Watson tree. The convergence theorem of Sah, Sahasrabudhe, and Sawhney, together with the Brown-measure identification following it, gives weak convergence of the empirical spectral measure of $A_n$ to $\mu_d$ in probability. We prove that the outer radius of its support is $\sqrt d$, using a resolvent recursion on Poisson--Galton--Watson trees.
We study the eigenvalues and eigenvectors of the adjacency matrix $A$ of the Erd\H{o}s-R\'enyi graph $\mathbb G(N,p)$ in the regime $Np \gg (\log N)^2$. We establish optimal isotropic delocalization for the bulk eigenvectors $\boldsymbol u$, meaning that $\langle \boldsymbol v, \boldsymbol u\rangle^2 \leq \frac{C \log...
Given an $n\times n$ symmetric matrix $M$ with largest eigenvalue $\lambda_1\geq 0$, it is easy to show that the solution of the optimisation problem $\max_{v\in [-1,1]^n}v^TMv$ is at most $\lambda_1 n$. We prove the following converse: if every $n'\times n'$ principal submatrix of $M$ has maximal eigenvalue at least $...
Oliver Janzer, István Tomon, Fredy Yip· 1 citation
For a graph $H$, let $\operatorname{ex}(Q_n, H)$ be the largest number of edges in a subgraph of the hypercube $Q_n$ of dimension $n$ that contains no subgraph isomorphic to $H$. The Tur\'an density of $H$ in a hypercube, denoted $\pi_\square(H)$, is defined as $\lim_{n\rightarrow \infty} \operatorname{ex}(Q_n, H)/|E(Q...
We consider a random subgraph $G_p(n,r,<s)$ of the complete distance graph $G(n,r,<s)$ whose vertices are the $r$-element subsets of the set $\{1,\dots,n\}$ and whose edges join pairs of subsets that intersect in fewer than $s$ elements; each edge survives independently of the others with probability $p$. The independe...
V. A. Pokhachevskiy, Andrei M. Raigorodskii· 2 citations
Let $A$ be the (non-symmetric) adjacency matrix of a uniformly random $d$-regular directed graph on $n$ vertices, and let $\Xi$ be independent of $A$ with i.i.d. Rademacher entries. Suppose that $\min(d,n-d)\ge\lambda n$ for some fixed $\lambda\in(0,1/2]$. We show that there exists $c>0$, depending only on $\lambda$, s...
Huai-Jin Liang, Yu-Xiao Shi, Ke-Xin Yu et al.· 1 citation
Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sa...
Asaf Cohen Antonir, Ilay Hoshen, M. Zhukovskii· 1 citation
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