论文标题
离散随机矩阵的奇异性
Singularity of discrete random matrices
论文作者
论文摘要
令$ξ$为具有有限支持的非恒定实价随机变量，让$ m_ {n}（ξ）$表示$ n \ times n $随机矩阵，带有$ξ$的独立副本的条目。对于$ξ$，在支持方面并不统一，我们表明\ begin {align*} \ mathbb {p} [m_ {n}（n}（ξ）（ξ）\ text {is sidular}]＆= \ m athbb {p}符号）行或列}]，\ end {align*}，从而确认了民间传说的猜想。作为特殊情况，我们获得： （1）对于$ξ= \ text {bernoulli}（p）$，带有固定$ p \ in（0,1/2）$，\ [\ mathbb {p} [m_ {n} [m_ {n}（ξ）（ξ）\ text {is singular} （1-p）^2）^{n}，\]，它确定了两个渐近术语的奇异性概率。以前，在随机矩阵的奇异性研究中没有这种精度的结果。 （2）对于$ξ= \ text {bernoulli}（p）$，带有固定的$ p \ in（1/2,1）$，\ [\ mathbb {p} [m_ {n} [m_ {n}（ξ）\ text {is singular}] =（1 + o_n（1 + o_n（1 + o_n（1 + o_n（1 + o_n（1 + o_n（n（1），N（1 + o_n（n（1））N（n-1）n（n-1）n（n-1）（p^2 +） $（\ sqrt {p} + o_n（1））^{n} $的上限较弱。 对于$ξ$，它的支持是统一的： （1）我们证明\ begin {align*} \ mathbb {p} [m_ {n}（ξ）\ text {is singular}]＆=（1+o_n（1））^{n} \ Mathbb {p} [p} [p} [\ text {\ text {两行或列是等于}]。 \ end {align*}（2）也许更重要的是，我们对单位球的“可压缩”部分的贡献进行了清晰的分析，该部分对$ m_ {n}（ξ）$的最小单数值的下尾的下尾。
Let $ξ$ be a non-constant real-valued random variable with finite support, and let $M_{n}(ξ)$ denote an $n\times n$ random matrix with entries that are independent copies of $ξ$. For $ξ$ which is not uniform on its support, we show that \begin{align*} \mathbb{P}[M_{n}(ξ)\text{ is singular}] &= \mathbb{P}[\text{zero row or column}] + (1+o_n(1))\mathbb{P}[\text{two equal (up to sign) rows or columns}], \end{align*} thereby confirming a folklore conjecture. As special cases, we obtain: (1) For $ξ= \text{Bernoulli}(p)$ with fixed $p \in (0,1/2)$, \[\mathbb{P}[M_{n}(ξ)\text{ is singular}] = 2n(1-p)^{n} + (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n},\] which determines the singularity probability to two asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. (2) For $ξ= \text{Bernoulli}(p)$ with fixed $p \in (1/2,1)$, \[\mathbb{P}[M_{n}(ξ)\text{ is singular}] = (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n}.\] Previously, only the much weaker upper bound of $(\sqrt{p} + o_n(1))^{n}$ was known due to the work of Bourgain-Vu-Wood. For $ξ$ which is uniform on its support: (1) We show that \begin{align*} \mathbb{P}[M_{n}(ξ)\text{ is singular}] &= (1+o_n(1))^{n}\mathbb{P}[\text{two rows or columns are equal}]. \end{align*} (2) Perhaps more importantly, we provide a sharp analysis of the contribution of the `compressible' part of the unit sphere to the lower tail of the smallest singular value of $M_{n}(ξ)$.
