论文标题
随机对称矩阵的最低奇异值
The least singular value of a random symmetric matrix
论文作者
论文摘要
令$ a $为$（a_ {i，j}）_ {i \ leq j} $的$ n \ times n $对称矩阵，独立且根据subgaussian分布相同分布。我们表明，$$ \ mathbb {p}（σ_ {\ min}（a）\ leq \ lepsilon/\ sqrt {n}）\ leq c \ leq c \ varepsilon + e^{ - cn}，$ q \ n where $ c _ {$σ_{\ a） $仅取决于$ a $的条目的分布。该结果证实了对随机对称矩阵最不奇异价值的低尾渐近造型的民间传说猜想，并且最好是常数对$ a_ {i，j} $的分布的依赖性。一路上，我们证明概率$ a $具有重复的特征值是$ e^{ - ω（n）} $，因此确认了Nguyen，Tao和Vu的猜想。
Let $A$ be a $n \times n$ symmetric matrix with $(A_{i,j})_{i\leq j} $, independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}(σ_{\min}(A) \leq \varepsilon/\sqrt{n}) \leq C \varepsilon + e^{-cn},$$ where $σ_{\min}(A)$ denotes the least singular value of $A$ and the constants $C,c>0 $ depend only on the distribution of the entries of $A$. This result confirms a folklore conjecture on the lower-tail asymptotics of the least singular value of random symmetric matrices and is best possible up to the dependence of the constants on the distribution of $A_{i,j}$. Along the way, we prove that the probability $A$ has a repeated eigenvalue is $e^{-Ω(n)}$, thus confirming a conjecture of Nguyen, Tao and Vu.
