AI 中文总结
研究随机符号矩阵为正态的概率，通过组合性证明得出其精确指数阶$Pr(M_nM_n^T = M_n^TM_n) = 2^{-N + O(n)}$，还表明随机$0 - 1$矩阵有相同概率，给出了该概率的精确指数渐近性。
AI 中文摘要
设$M_n$是一个$n\times n$的随机矩阵，其元素为独立的拉德马赫随机变量，令$N = \binom{n}{2}$。我们证明\[ Pr(M_nM_n^T = M_n^TM_n) = 2^{-N + O(n)} \]。这给出了随机符号矩阵为正态的概率的精确指数阶。下界由对称符号矩阵给出。我们还记录了随机$0 - 1$矩阵具有相同精确指数正态概率的直接结果。匹配上界的证明是组合性的：在对非对角元素的对称/反对称类型模式进行条件设定后，模4约简给出了一个在$F_2$上的线性方程组；秩对偶论证将类型模式的求和转化为$F_2$上可交换对称对的计数；并且通过使用平衡对称双线性形式和标准有限域中心化子公式对有理标准型进行求和来界定这个计数。
英文摘要
Let $M_n$ be an $n\times n$ random matrix whose entries are independent Rademacher random variables, and put $N=\binom n2$. We prove \[ Pr(M_nM_n^T=M_n^TM_n)=2^{-N+O(n)}. \] This gives the sharp exponential order for the probability that a random sign matrix is normal. The lower bound is supplied by symmetric sign matrices. We also record the immediate consequence that random $0$-$1$ matrices have the same sharp exponential normality probability. The proof of the matching upper bound is combinatorial: after conditioning on the symmetric/skew-symmetric type pattern of the off-diagonal entries, a mod-$4$ reduction gives a system of linear equations over $F_2$; a rank-duality argument converts the sum over type patterns into a count of commuting symmetric pairs over $F_2$; and this count is bounded by summing over rational canonical types, using balanced symmetric bilinear forms and the standard finite-field centralizer formula.
