Fluctuations of the quadratic matching cost on the flat torus: dimensions five and higher
Abstract
We prove a central limit theorem for the quadratic matching cost between the uniform measure and its empirical counterpart on the flat torus in dimensions d≥5. Let μ be Haar probability measure on 𝕋d and let μn be the empirical measure of n independent samples from μ. Then (W22(μn,μ)−𝔼W22(μn,μ))/√(Var(W22(μn,μ))) &Longrightarrow N(0,1), cdn−1−4/d ≤ Var(W22(μn,μ)) &≤ Cdn−1−4/d, for positive constants cd,Cd depending only on d and all sufficiently large n. The proof introduces a random-walk approach to matching fluctuations. We construct a random walk that jumps between neighboring cells of the optimal partition and use it to describe how the matching cost responds to changes in the sample. Estimates for this walk control the effect of replacing one sample point and the interaction between two replacements. These bounds yield the variance estimates and allow us to apply a martingale central limit theorem.
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BibTeX
@misc{feng2026fluctuations,
title = {{Fluctuations of the quadratic matching cost on the flat torus: dimensions five and higher}},
author = {Shi Feng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14304},
eprint = {2610.14304},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14304},
}
RIS
TY - UNPB
TI - Fluctuations of the quadratic matching cost on the flat torus: dimensions five and higher
AU - Shi Feng
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14304
UR - https://arxiv.org/abs/2610.14304
AB - We prove a central limit theorem for the quadratic matching cost between the uniform measure and its empirical counterpart on the flat torus in dimensions d≥5. Let μ be Haar probability measure on 𝕋^d and let μ_n be the empirical measure of n independent samples from μ. Then (W_2^2(μ_n,μ)−𝔼W_2^2(μ_n,μ))/√(Var(W_2^2(μ_n,μ))) &Longrightarrow N(0,1), c_dn^−1−4/d ≤ Var(W_2^2(μ_n,μ)) &≤ C_dn^−1−4/d, for positive constants c_d,C_d depending only on d and all sufficiently large n. The proof introduces a random-walk approach to matching fluctuations. We construct a random walk that jumps between neighboring cells of the optimal partition and use it to describe how the matching cost responds to changes in the sample. Estimates for this walk control the effect of replacing one sample point and the interaction between two replacements. These bounds yield the variance estimates and allow us to apply a martingale central limit theorem.
ER -