Singularity of sparse random matrices: simple proofs. (15th January 2022) - Ferber, Asaf; Kwan, Matthew; Sauermann, Lisa
Abstract: Consider a random $n× n$ zero-one matrix with 'sparsity' p, sampled according to one of the following two models: either every entry is independently taken to be one with probability p (the 'Bernoulli' model) or each row is independently uniformly sampled from the set of all length- n zero-one vectors with exactly pn ones (the 'combinatorial' model). We give simple proofs of the (essentially best-possible) fact that in both models, if $\min(p, 1-p)≥ (1+ε)\log n/n$ for any constant $ε>0$, then our random matrix is nonsingular with probability $1-o(1)$ . In the Bernoulli model, this fact was already well known, but in the combinatorial model this resolves a conjecture of Aigner-Horev and Person.