Singularity of sparse random matrices: simple proofs

Consider a random $n\times n$ 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 $\min(p,1-p)\geq (1+\varepsilon)\log n/n$ for any constant $\varepsilon>0$ , 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.