The smallest singular value of sparse discrete random matrices

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 $\eta \geq 0$, $\textbf{P}\left(s_n\left(M_n\right) \leq \eta\right) \lesssim n^C \eta+\exp \left(-n^c\right)$. As an application, we give an approximate Spielman-Teng theorem for $M_n$ whose entries are $\mu-$lazy random variables, extending previous work by Tao and Vu.