随机置换矩阵以高概率构成一组基
Random Permutation Matrices Form a Basis with High Probability
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中文总结 AI 辅助
本文证明随机选取的置换矩阵以高概率线性无关，证实了相关猜想，并给出了均匀子集模型的下界。
中文摘要 AI 辅助
设 $d_n=(n-1)^2+1$，即 $n\times n$ 置换矩阵的实线性张成的维数。我们证明 $d_n$ 个独立均匀随机置换矩阵以概率 $1-O(n^{-1/2})$ 线性无关。在给定互异的条件下，对于均匀随机的 $d_n$ 元子集也能得到相同结论，从而证实了 Kushwaha 和 Tripathi 的一个猜想。证明结合了三个要素：赋值泛函的模 2 复杂度参数、Roos 的特征函数估计（以 Do--Nguyen--Phan--Tran--Vu 记录的形式），以及 Ferber--Kwan--Sauermann 的核分解论证。对于均匀子集模型，我们还记录了由未占据矩阵位置得到的初等下界 $\exp(3/2+o(1))n^2e^{-n}$。
英文摘要
Let $d=(n-1)^2+1$, the dimension of the real linear span of the $n\times n$ permutation matrices. We prove that $d$ independent uniformly random permutation matrices fail to form a basis with probability $(1+o(1))n^2(1-1/n)^d=(e^{3/2}+o(1))n^2e^{-n}$. The same asymptotic holds for a uniformly random $d$-element subset, confirming a conjecture of Kushwaha and Tripathi and identifying the leading obstruction: a matrix position avoided by every sample. More generally, for any fixed number of additional samples, we determine the first three exponential orders of the failure probability. To prove these results, we develop support estimates valid over arbitrary fields, derive Fourier bounds from permanental minors, and introduce a counting argument for concentrated assignment functionals over large prime fields. We also give a sparse lifting argument showing that real rank deficiency without an annihilating functional of small support has probability $o(e^{-Cn})$ for every fixed $C>0$.
发表机构
- School of Mathematical Sciences, Zhejiang University（浙江大学本科数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
