Singularity of sparse random matrices: simple proofs

Abstract 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 be probability (the ‘Bernoulli’ model) or each row uniformly from set all length- n vectors exactly pn ones ‘combinatorial’ model). We give simple proofs (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 nonsingular $1-o(1)$ . In Bernoulli model, this was already well known, but combinatorial model resolves conjecture Aigner-Horev and Person.