The smallest singular value of sparse discrete random matrices

Abstract

Let $M_n$ be an $n\times n$ random matrix with i.i.d. sparse discrete entries. In this paper, we develop a simple framework to solve the approximate Spielman-Teng theorem for $M_n$, which has the following form: There exist constants $C,c>0$ such that for all $\hat{I}\cdot \ge 0$, $\text{P}\left(s_n\left(M_n\right)\le \hat{I}\cdot \right)\lesssim n^C\hat{I}\cdot +\exp \left(-n^c\right)$. As an application, we give an approximate Spielman-Teng theorem for $M_n$ whose entries are $\hat{I}¼-$lazy random variables, extending previous work by Tao and Vu.