平坦环面上二次匹配代价的波动：维度二、三和四
Fluctuations of the quadratic matching cost on the flat torus: dimensions two, three and four
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- Cornell University（康奈尔大学）
- Yale University（耶鲁大学）
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中文总结 AI 辅助
本文研究平坦环面维度2、3、4上二次最优匹配代价的方差渐近与极限分布，利用Hoeffding分解证明其等价于二阶U-统计量，得到显式方差常数及非高斯极限。
中文摘要 AI 辅助
我们建立了在平坦环面 $\mathbb T^d$ 上，维度 $d=2,3,4$ 时，均匀测度与其经验测度对应物之间的二次最优匹配代价的尖锐方差渐近性和极限分布。更精确地，设 $\mu$ 为 Haar 概率测度，$\mu_n$ 为从 $\mu$ 中抽取的 $n$ 个独立样本的经验测度。那么，对于 $d=2,3$，$$ \begin{aligned} n\bigl(W_2^2(\mu_n,\mu)-\mathbb E[W_2^2(\mu_n,\mu)]\bigr) &\xrightarrow{\mathrm d} L_d,\\\\ n^2\operatorname{Var}(W_2^2(\mu_n,\mu)) &\longrightarrow \frac{1}{8\pi^4} \sum_{k\in\mathbb Z^d\setminus\{0\}}\frac{1}{|k|^4}, \end{aligned} $$ 其中 $L_d$ 是独立的中心化指数随机变量的显式非高斯加权和。在维度四中，$$ \frac{n}{\sqrt{\log n}} \bigl(W_2^2(\mu_n,\mu)-\mathbb E[W_2^2(\mu_n,\mu)]\bigr) \xrightarrow{\mathrm d} N\left(0,\frac{1}{16\pi^2}\right). $$ 证明的主要思想是依赖于最优传输代价的 Hoeffding 分解，该分解在渐近上等价于二阶 U-统计量。
英文摘要
We establish sharp variance asymptotics and limiting distributions for the quadratic optimal matching cost between the uniform measure and an empirical measure counterpart on the flat torus $\mathbb T^d$ in dimensions $d=2,3,4$. More precisely, let $μ$ be Haar probability measure and $μ_n$ the empirical measure of $n$ independent samples drawn from $μ$. Then, for $d=2,3$, $$ \begin{aligned} n\bigl(W_2^2(μ_n,μ)-\mathbb E[W_2^2(μ_n,μ)]\bigr) &\xrightarrow{\mathrm d} L_d,\\ n^2\operatorname{Var}(W_2^2(μ_n,μ)) &\longrightarrow \frac{1}{8π^4} \sum_{k\in\mathbb Z^d\setminus\{0\}}\frac{1}{|k|^4}, \end{aligned} $$ where $L_d$ is an explicit non-Gaussian weighted sum of independent centered exponential random variables. In dimension four, $$ \frac{n}{\sqrt{\log n}} \bigl(W_2^2(μ_n,μ)-\mathbb E[W_2^2(μ_n,μ)]\bigr) \xrightarrow{\mathrm d} N\left(0,\frac{1}{16π^2}\right). $$ The main idea of the proof is to rely on a Hoeffding decomposition of the optimal transport cost, which turns out to be asymptotically equivalent to U-statistics of order 2.
