Fluctuations of the quadratic matching cost on the flat torus: dimensions two, three and four
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Feng, Shi, Mordant, Gilles
Institutions: Cornell University, Yale University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214088492