论文标题
随机Bernoulli矩阵的尖锐可逆性
Sharp invertibility of random Bernoulli matrices
论文作者
论文摘要
令$ p \ in（0,1/2）$修复，然后让$ b_n（p）$为a $ n \ times n $随机矩阵，随机矩阵带有i.i.d. Bernoulli随机变量具有平均$ P $。我们证明了所有$ t \ ge 0 $，\ [\ mathbb {p} [s_n（b_n（p））\ le tn^{ - 1/2}] \ le c_p t + c_p t + 2n（1-p）^{n} $ b_n（p）$和$ c_p，ε_p> 0 $仅取决于$ p $。特别是，\ [\ mathbb {p} [b_ {n}（p）\ text {singular}] = 2n（1-p）^{n} + c_ {p}（1-p-ε_p）^{n}，\]，证实了litvak和tikhomirov的猜测。 我们还通过证明$ q_ {n} $是一个$ n \ times n $随机矩阵，其独立行分布在$ \ {0,1 \}^{n} $的中央切片上，然后\ [Q _ p} { o_n（1））^{n}。\ \]这首先提供了对具有依赖条目的随机离散矩阵的任何自然模型中奇异性概率的对数。
Let $p \in (0,1/2)$ be fixed, and let $B_n(p)$ be an $n\times n$ random matrix with i.i.d. Bernoulli random variables with mean $p$. We show that for all $t \ge 0$, \[\mathbb{P}[s_n(B_n(p)) \le tn^{-1/2}] \le C_p t + 2n(1-p)^{n} + C_p (1-p-ε_p)^{n},\] where $s_n(B_n(p))$ denotes the least singular value of $B_n(p)$ and $C_p, ε_p > 0$ are constants depending only on $p$. In particular, \[\mathbb{P}[B_{n}(p) \text{ is singular}] = 2n(1-p)^{n} + C_{p}(1-p-ε_p)^{n},\] which confirms a conjecture of Litvak and Tikhomirov. We also confirm a conjecture of Nguyen by showing that if $Q_{n}$ is an $n\times n$ random matrix with independent rows that are uniformly distributed on the central slice of $\{0,1\}^{n}$, then \[\mathbb{P}[Q_{n} \text{ is singular}] = (1/2 + o_n(1))^{n}.\] This provides, for the first time, a sharp determination of the logarithm of the probability of singularity in any natural model of random discrete matrices with dependent entries.
