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.

中文翻译：

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稀疏随机矩阵的奇点：简单证明

考虑随机$n\次 n$具有“稀疏性”的零一矩阵p，根据以下两种模型之一进行采样：或者每个条目都被独立地视为具有概率的条目p（“伯努利”模型）或每一行独立地从所有长度的集合中均匀采样 -n零一向量正好PN那些（“组合”模型）。我们给出了（基本上是最好的）事实的简单证明，在这两个模型中，如果$\min(p,1-p)\geq (1+\varepsilon)\log n/n$对于任何常数$\伐普西隆>0$, 那么我们的随机矩阵是非奇异的概率$1-o(1)$. 在伯努利模型中，这一事实已经众所周知，但在组合模型中，这解决了 Aigner-Horev 和 Person 的猜想。