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 P [det(Q(n,d)) = 0] = O (n(1/2) log(3/2) n/d) = o(1), whenever omega(n(1/2) log(3/2) n) = d < n/2. d 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 P [det(A + Qn,d) = 0] = ) O (n(1/2) log(3/2)n, again, whenever omega(n(1/2)log(3/2)n) = d < n/2 and A has the property that d (1, -d) is not an eigenpair of A.(c) 2022 Elsevier B.V. All rights reserved.