AI 中文总结
本文证明了奇特征有限域上随机矩阵积和式的渐近均匀分布猜想，给出了收敛速度的一致上界，且该结论对随n变化的奇素数幂序列同样成立。
AI 中文摘要
设$q$为奇素数幂，令$A_n=(a_{ij})\in\mathbb F_q^{n\times n}$为元素独立且均匀分布在$\mathbb F_q$上的随机矩阵。$A_n$的积和式定义为$\operatorname{per}(A_n)=\sum_{σ\in S_n}\prod_{i=1}^n a_{i,σ(i)}$，其中$S_n$表示$[n]$上的对称群。Ghasemi、Gross与Kopparty提出猜想：对任意固定奇素数幂$q$，零质量渐近性质$\Pr[\operatorname{per}(A_n)=0]=1/q+o(1)$成立；随后Hunter、Kwan与Sauermann给出了等价的全分布表述：对任意固定$q$和任意$x\in\mathbb F_q$，有$\lim_{n\to\infty}\Pr[\operatorname{per}(A_n)=x]=\frac1q$。本文证明了该猜想。更确切地说，本文证明存在绝对常数$C>0$，使得对任意奇素数幂$q$和任意$n\ge 7$，有$\frac12\sum_{x\in\mathbb F_q}\left|\Pr[\operatorname{per}(A_n)=x]-\frac1q\right|\le C\frac{\log n}{n}$。该估计对$q$一致成立，因此结论对任意奇素数幂序列$q=q(n)$均有效。
英文摘要
Let $q$ be an odd prime power, and let $A_n=(a_{ij})\in\mathbb F_q^{n\times n}$ be a random matrix whose entries are independent and uniformly distributed on $\mathbb F_q$. The permanent of $A_n$ is defined by $\operatorname{per}(A_n)=\sum_{σ\in S_n}\prod_{i=1}^n a_{i,σ(i)}$, where $S_n$ denotes the symmetric group on $[n]$. Ghasemi, Gross, and Kopparty conjectured the zero-mass asymptotic $\Pr[\operatorname{per}(A_n)=0]=1/q+o(1)$ for every fixed odd prime power $q$, and Hunter, Kwan, and Sauermann subsequently stated its equivalent full-distribution formulation: for every fixed $q$ and every $x\in\mathbb F_q$, \[ \lim_{n\to\infty}\Pr[\operatorname{per}(A_n)=x]=\frac1q. \] In this paper, we prove this conjecture. More precisely, we prove that there is an absolute constant $C>0$ such that \[\frac12\sum_{x\in\mathbb F_q}\left|\Pr[\operatorname{per}(A_n)=x]-\frac1q\right|\le C\frac{\log n}{n}\] for every odd prime power $q$ and every $n\ge 7$. The estimate is uniform in $q$, so the conclusion remains valid for every sequence $q=q(n)$ of odd prime powers.
Comments13 pages. Comments welcome. This version adds a Declaration on the Use of AI; the mathematical content is unchanged
