On sparse random combinatorial matrices

Let Q_n,d denote the random combinatorial matrix whose rows are independent of one another and such that each row is sampled uniformly at random from the subset of vectors in {0,1}ⁿ having precisely d entries equal to 1. We present a short proof of the fact that [Formula presented], whenever ω(n^1/2log^3/2⁡n)=d≤n/2. In particular, our proof accommodates sparse random combinatorial matrices in the sense that d=o(n) is allowed. We also consider the singularity of deterministic integer matrices A randomly perturbed by a sparse combinatorial matrix. In particular, we prove that [Formula presented], again, whenever ω(n^1/2log^3/2⁡n)=d≤n/2 and A has the property that (1,−d) is not an eigenpair of A.