Let \(\xi \) be a non-constant real-valued random variable with finite support and let \(M_{n}(\xi )\) denote an \(n\times n\) random matrix with entries that are independent copies of \(\xi \). For \(\xi \) which is not uniform on its support, we show that

thereby confirming a folklore conjecture. As special cases, we obtain:

• For \(\xi = {\text {Bernoulli}}(p)\) with fixed \(p \in (0,1/2)\),

  $$\begin{aligned} {\mathbb {P}}[M_{n}(\xi )\text { is singular}] = 2n(1-p)^{n} + (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n}, \end{aligned}$$

  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. The first asymptotic term confirms a conjecture of Litvak and Tikhomirov.

• For \(\xi = {\text {Bernoulli}}(p)\) with fixed \(p \in (1/2,1)\),

  $$\begin{aligned} {\mathbb {P}}[M_{n}(\xi )\text { is singular}] = (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n}. \end{aligned}$$

  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 \(\xi \) which is uniform on its support:

• We show that

  $$\begin{aligned} {\mathbb {P}}[M_{n}(\xi )\text { is singular}]&= (1+o_n(1))^{n}{\mathbb {P}}[\text {two rows or columns are equal}]. \end{aligned}$$

• 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 \(M_{n}(\xi )\).

令\(\xi \)是一个具有有限支持的非常量实值随机变量，并令\(M_{n}(\xi )\)表示一个\(n\times n\)随机矩阵，其条目为\(\xi \) 的独立副本。对于\(\xi \)在其支持度上不均匀，我们证明

从而证实了民间传说的猜想。作为特殊情况，我们获得：

• 对于\(\xi = {\text {Bernoulli}}(p)\)与固定\(p \in (0,1/2)\)，

  $$\begin{aligned} {\mathbb {P}}[M_{n}(\xi )\text { 是单数}] = 2n(1-p)^{n} + (1+o_n(1)) n(n-1)(p^2 + (1-p)^2)^{n}, \end{aligned}$$

  它确定了两个渐近项的奇异概率。以前，在随机矩阵奇异性的研究中，没有这样精确的结果。第一个渐近项证实了 Litvak 和 Tikhomirov 的猜想。

• 对于\(\xi = {\text {Bernoulli}}(p)\)与固定\(p \in (1/2,1)\)，

  $$\begin{aligned} {\mathbb {P}}[M_{n}(\xi )\text { 是单数}] = (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n}。\end{对齐}$$

  以前，由于 Bourgain-Vu-Wood 的工作，只知道\((\sqrt{p} + o_n(1))^{n}\) 的弱得多的上限。

对于\(\xi \)在其支持上是统一的：

• 我们证明

  $$\begin{aligned} {\mathbb {P}}[M_{n}(\xi )\text { 是单数}]&= (1+o_n(1))^{n}{\mathbb {P} }[\text {两行或两列相等}]。\end{对齐}$$

• 也许更重要的是，我们对单位球体的“可压缩”部分对\(M_{n}(\xi )\)的最小奇异值的下尾的贡献进行了清晰的分析。