On the singularity of random combinatorial matrices

TL;DR

This paper proves that a certain class of random (0,1) matrices with fixed row sums are almost surely non-singular as the size grows, using a novel inverse Littlewood-Offord approach.

Contribution

It introduces a new inverse Littlewood-Offord technique to analyze the singularity probability of fixed-row-sum random matrices.

Findings

The probability of singularity decreases faster than any polynomial in matrix size.

Random matrices with fixed row sums are almost surely invertible for large sizes.

The method can be applied to other structured random matrices.

Abstract

It is shown that a random $(0,1)$ matrix whose rows are independent random vectors of exactly $n/2$ zero components is non-singular with probability $1-O(n^{-C})$ for any $C>0$. The proof uses a non-standard inverse-type Littlewood-Offord result.