# Sharp transition of the invertibility of the adjacency matrices of sparse random graphs

@article{Basak2018SharpTO,
title={Sharp transition of the invertibility of the adjacency matrices of sparse random graphs},
author={Anirban Basak and Mark Rudelson},
journal={Probability Theory and Related Fields},
year={2018},
volume={180},
pages={233 - 308}
}
• Published 22 September 2018
• Mathematics
• Probability Theory and Related Fields
We consider three models of sparse random graphs: undirected and directed Erdős–Rényi graphs and random bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies np≥logn+k(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$np\ge \log n+k(n)$$\end{document…
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Geometric and Functional Analysis
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The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized $${n \times n}$$n×n matrix with i.i.d. entries converges to the uniform measure on the unit disc as
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Let (Xjk)jk≥1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ2. Let M be the n × n random Markov matrix with i.i.d. rows defined by
In this paper, we investigate the invertibility of sparse symmetric matrices. We will show that for an $n\times n$ sparse symmetric random matrix $A$ with $A_{ij} = \delta_{ij} \xi_{ij}$ is
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For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and
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