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}
}
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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