Exponential Rank Bounds for Random Matrices

Abstract

Fix $b\in(0,1)$, let $1\leq k\leq n$, and let $A=(A_{ij})$ be an $n\times n$ random matrix with independent real entries satisfying $$ \sup_{x\in\mathbb{R}}\mathbb{P}\{A_{ij}=x\}\leq b<1 \qquad (1\leq i,j\leq n). $$ We show that there exists $c>0$ such that $$ \mathbb{P}\{\operatorname{rank} A\leq n-k\}\leq \exp(-cnk), \qquad 1\leq k\leq n. $$