Related papers: Local law and complete eigenvector delocalization …

We study the graph alignment problem over two independent Erd\H{o}s-R\'enyi graphs on $n$ vertices, with edge density $p$ falling into two regimes separated by the critical window around $p_c=\sqrt{\log n/n}$. Our result reveals an…

For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal value of various problems. However, there has not been as much success in…

A graph $G$ is said to be $p$-locally dense if every induced subgraph of $G$ with linearly many vertices has edge density at least $p$. A famous conjecture of Kohayakawa, Nagle, R\"odl, and Schacht predicts that locally dense graphs have,…

We show that if $p=O(1/n)$, then the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ with high probability admits a canonical labeling computable in time $O(n\log n)$. Combined with the previous results on the canonization of random graphs, this…

Let $\mathcal{G}(N,\frac 1Nt_N)$ be the Erd\H{o}s-R\'enyi graph with connection probability $\frac 1Nt_N\sim t/N$ as $N\to\infty$ for a fixed $t\in(0,\infty)$. We derive a large-deviations principle for the empirical measure of the sizes of…

The local complement G*i of a simple graph G at one of its vertices i is obtained by complementing the subgraph induced by the neighborhood of i and leaving the rest of the graph unchanged. If e={i,j} is an edge of G then G*e=((G*i)*j)*i is…

We construct a derived variant of Emerton's eigenvarieties using the locally analytic representation theory of $p$-adic groups. The main innovations include comparison and exploitation of two homotopy equivalent completed complexes…

This article is concerned with properties of delocalization for eigenfunctions of Schr\"odinger operators on large finite graphs. More specifically, we show that the eigenfunctions have a large support and we assess their lp-norms. Our…

We analyse the eigenvalues of Erd\"os--R\'enyi random bipartite graphs. In particular, we consider $p$ satisfying $n_{1}p=\Omega(\sqrt{n_{1}p}\log^{3}(n_{1})),$ $n_{2}p=\Omega(\sqrt{n_{2}p}\log^{3}(n_{2})),$ and let $G\sim…

We consider a Gaussian random matrix with correlated entries that have a power law decay of order $d>2$ and prove universality for the extreme eigenvalues. A local law is proved using the self-consistent equation combined with a…

In this note we consider a more general version of local sparsity introduced recently by Anderson, Kuchukova, and the author. In particular, we say a graph $G = (V, E)$ is $(k, r)$-locally-sparse if for each vertex $v \in V(G)$, the…

Let $d\geq 3$ be a fixed integer, and a prime number $p$ such that $\gcd(p,d)=1$. Let $A$ be the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. We show that as a random matrix in ${\mathbb F}_p$, \begin{equation}…

We connect the mixing behaviour of random walks over a graph to the power of the local-consistency algorithm for the solution of the corresponding constraint satisfaction problem (CSP). We extend this connection to arbitrary CSPs and their…

We consider deformed sparse random matrices of the form $H= W+ \lambda V$, where $W$ is a real symmetric sparse random matrix, $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$, and $\lambda = O(1)…

We investigate the problem of detecting correlation between two Erd\H{o}s-R\'enyi graphs $G(n,p)$, formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent, while under the alternative…

We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying…

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erd\H{o}s, \L uczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a…

We study the spectral properties of sparse random graphs with different topologies and type of interactions, and their implications on the stability of complex systems, with particular attention to ecosystems. Specifically, we focus on the…

These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero…

We consider products of independent square non-Hermitian random matrices. More precisely, let X(1),...,X(n) be random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance…