Fluctuations of the quadratic matching cost on the flat torus: dimensions two, three and four
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 𝕋d in dimensions d=2,3,4. More precisely, let μ be Haar probability measure and μn the empirical measure of n independent samples drawn from μ. Then, for d=2,3, n(W22(μn,μ)−𝔼[W22(μn,μ)]) &xrightarrowd Ld, n2Var(W22(μn,μ)) &⟶ 1/8π4 ∑k∈ℤd∖{0}1/|k|4, where Ld is an explicit non-Gaussian weighted sum of independent centered exponential random variables. In dimension four, n/(√(log n)) (W22(μn,μ)−𝔼[W22(μn,μ)]) xrightarrowd N(0,1/16π2). 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.
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BibTeX
@misc{feng2026fluctuations,
title = {{Fluctuations of the quadratic matching cost on the flat torus: dimensions two, three and four}},
author = {Shi Feng and Gilles Mordant},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26597},
eprint = {2609.26597},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26597},
}
RIS
TY - UNPB
TI - Fluctuations of the quadratic matching cost on the flat torus: dimensions two, three and four
AU - Shi Feng
AU - Gilles Mordant
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26597
UR - https://arxiv.org/abs/2609.26597
AB - 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 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, n(W_2^2(μ_n,μ)−𝔼[W_2^2(μ_n,μ)]) &xrightarrowd L_d, n^2Var(W_2^2(μ_n,μ)) &⟶ 1/8π^4 ∑_k∈ℤ^d∖{0}1/|k|^4, where L_d is an explicit non-Gaussian weighted sum of independent centered exponential random variables. In dimension four, n/(√(log n)) (W_2^2(μ_n,μ)−𝔼[W_2^2(μ_n,μ)]) xrightarrowd N(0,1/16π^2). 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.
ER -