The smallest singular value of random combinatorial matrices (2007.06318v5)

Abstract: Let $Q_n$ be a random $n\times n$ matrix with entries in ${0,1}$ whose rows are independent vectors of exactly $n/2$ zero components. We show that the smallest singular value $s_n(Q_n)$ of $Q_n$ satisfies [ \mathbb{P}\Big{s_n(Q_n)\le \frac{\varepsilon}{\sqrt{n}}\Big} \le C\varepsilon + 2 e^{-cn} \quad \forall \varepsilon \ge 0, ] which is optimal up to the constants $C,c>0$. This improves on earlier results of Ferber, Jain, Luh and Samotij, as well as Jain. In particular, for $\varepsilon=0$, we obtain the first exponential bound in dimension for the singularity probability [ \mathbb{P}\big{Q_n \,\,\text{is singular}\big} \le 2 e^{-cn}.] To overcome the lack of independence between entries of $Q_n$, we introduce an arithmetic-combinatorial invariant of a pair of vectors, which we call a Combinatorial Least Common Denominator (CLCD). We prove a small ball probability inequality for the combinatorial statistic $\sum_{i=1}^{n}a_iv_{\sigma(i)}$ in terms of the CLCD of the pair $(a,v)$, where $\sigma$ is a uniformly random permutation of ${1,2,\ldots,n}$ and $a:=(a_1,\ldots,a_n), v:=(v_1,\ldots,v_n)$ are real vectors. This inequality allows us to derive strong anti-concentration properties for the distance between a fixed row of $Q_n$ and the linear space spanned by the remaining rows, and prove the main result.