论文标题
稀疏的伯努利矩阵的奇异性
Singularity of sparse Bernoulli matrices
论文作者
论文摘要
令$ m_n $为$ n \ times n $随机矩阵，带有i.i.d。 Bernoulli（P）条目。我们表明，有一个通用常数$ c \ geq 1 $，这样，每当$ p $和$ n $满意时，$ c \ log n/n \ leq p \ leq p \ leq c^{ - 1} $，\ begin {align*} {\ mathbb p} {\ mathbb p}单数} \ big \}＆=（1+o_n（1））{\ mathbb p} \ big \ {\ mbox {$ m_n $包含一个零行或列} \ big \} \} \\＆=（2+o_n（2+o_n（2+o_n（1））数量将零作为$ n \ to \ infty $收敛。我们在$ m_n $的最小单数值上提供相应的上限和下限。
Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$, \begin{align*} {\mathbb P}\big\{\mbox{$M_n$ is singular}\big\}&=(1+o_n(1)){\mathbb P}\big\{\mbox{$M_n$ contains a zero row or column}\big\}\\ &=(2+o_n(1))n\,(1-p)^n, \end{align*} where $o_n(1)$ denotes a quantity which converges to zero as $n\to\infty$. We provide the corresponding upper and lower bounds on the smallest singular value of $M_n$ as well.
