The smallest singular value of random combinatorial matrices

TL;DR

This paper establishes optimal bounds on the smallest singular value and the probability of singularity for random combinatorial matrices with fixed row sums, introducing a new combinatorial invariant to handle dependencies.

Contribution

It introduces the Combinatorial Least Common Denominator (CLCD) to analyze dependencies and proves exponential bounds on singularity probability for such matrices.

Findings

Bound on smallest singular value of $Q_n$ with high probability

First exponential bound on singularity probability for combinatorial matrices

Introduction of CLCD for anti-concentration analysis

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.