精确环条件约束下的谱极值
Spectral extremes under exact cycle conditioning
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中文总结 AI 辅助
本文证明随机置换谱极大值的对数波动完全由环数决定，二者渐近完全对齐，并给出精确条件分布下的统一极限定理。
中文摘要 AI 辅助
设 $P_n$ 为 $n$ 个符号的随机置换矩阵，并设 $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$。Cook 和 Zeitouni 证明了对于均匀置换，$M_n/\log n$ 依概率收敛于常数 $x_0$。我们证明 $M_n$ 的 $\sqrt{\log n}$ 阶波动完全由环数 $K_n$ 承载。记 $\lambda(s)=\log\{\Gamma(1+s)/\Gamma(1+s/2)^2\}$，令 $s_\kappa$ 在 $(0,\infty)$ 上最小化 $(1+\kappa\lambda(s))/s$，并令 $v(\kappa)=\kappa\lambda'(s_\kappa)$ 和 $a_\theta=\lambda(s_\theta)/s_\theta$。在参数 $\theta>0$ 固定的 Ewens 测度下，我们证明 $M_n=v(\theta)\log n+a_\theta(K_n-\theta\log n)+O_P(\log\log n)$，因此标准化对 $(K_n,M_n)$ 联合收敛于 $(G,G)$，其中 $G$ 为标准正态：最大值与环数在渐近意义下完全对齐。这是从关于精确条件分布的命题推导出的，该命题不依赖于 $\theta$：对每个紧区间 $[\kappa_-,\kappa_+]\subset(0,\infty)$，存在有限常数 $C$，使得 $P(|M_n-v(k/\log n)\log n|>C\log\log n \mid K_n=k)$ 对满足 $\kappa_-\log n\le k\le\kappa_+\log n$ 的所有整数 $k$ 一致地趋于 0，即对精确且可能非典型的环数成立。证明在双变量系数提取中同时保持大小和环数。长度超过 $n/(\log n)^4$ 的环被保留为解析因子，其系数在较短环产生的每个大小平移下都是平坦的；正性将标量系数渐近转化为整个路径约束测度的相对比较，且误差不随约束数量或事件的稀有性而恶化。约束下界来自沿二进端点盒链的 killed 卷积的逐点鞍点估计。
英文摘要
Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cycles $K_n$. Write $λ(s)=\log\{Γ(1+s)/Γ(1+s/2)^2\}$, let $s_κ$ minimize $(1+κλ(s))/s$ on $(0,\infty)$, and put $v(κ)=κλ'(s_κ)$ and $a_θ=λ(s_θ)/s_θ$. Under the Ewens measure with any fixed parameter $θ>0$ we prove $M_n=v(θ)\log n+a_θ(K_n-θ\log n)+O_P(\log\log n)$, so that the standardized pair $(K_n,M_n)$ converges jointly to $(G,G)$ with $G$ standard normal: the maximum and the cycle count are asymptotically perfectly aligned. This is deduced from a statement about the exact conditional law, which does not depend on $θ$: for every compact $[κ_-,κ_+]\subset(0,\infty)$ there is a finite $C$ such that $P(|M_n-v(k/\log n)\log n|>C\log\log n \mid K_n=k)$ tends to $0$ uniformly over integers $k$ with $κ_-\log n\le k\leκ_+\log n$, that is, over exact and possibly atypical cycle counts. The proof keeps the size and the cycle count simultaneously in a two-variable coefficient extraction. Cycles longer than $n/(\log n)^4$ are reserved as an analytic factor whose coefficients are flat under every size shift produced by the shorter cycles; positivity then converts a scalar coefficient asymptotic into a relative comparison of the entire path-constrained measure, with an error that does not degrade with the number of constraints or with the rarity of the event. The constrained lower bound comes from pointwise saddle estimates for killed convolutions along a dyadic chain of endpoint boxes.
发表机构
- Shenzhen MSU–BIT University（深圳莫斯科国立技术大学）
- Guangdong Laboratory of Machine Perception and Intelligent Computing（广东省机器感知与智能计算实验室）
机构由 AI 辅助整理，请以论文原文为准。
