The smallest singular value of sparse discrete random matrices

Let Mₙ be an n × 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 Mₙ, which has the following form: There exist constants $C, c>0$ such that for all η≥ 0, P(sₙ(Mₙ) ≤ η) nC η+exp (-nᶜ). As an application, we give an approximate Spielman-Teng theorem for Mₙ whose entries are $μ-$lazy random variables, extending previous work by Tao and Vu.