The smallest singular value of signed random combinatorial matrices

Abstract

Let $M_n$ be an $n\times n$ signed random combinatorial matrix whose rows are independent and uniformly distributed over the set of $\{-1,0,1\}$-vectors with exactly $n/2$ zero coordinates. Despite the dependence induced by the row constraints, we prove that there exist constants $C,c>0$ such that for any $\epsilon \ge 0$, \begin{align*} \textbf{P}\left(s_{n}(M_n)\le {\varepsilon}{n^{-1/2}}\right)\le C\varepsilon+e^{-cn}. \end{align*} In particular, the probability that $M_n$ is singular is exponentially small. Our approach builds on the Combinatorial Least Common Denominator (CLCD) introduced by Tran and develops the method in the present constrained setting.