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 $\eta \ge 0$, $\mathbf{\text{P}}(s_n\left(M_n\right)\le \eta )\lesssim n^C\eta +\exp (-n^c)$. 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.