{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26597",
  "author_total": 2,
  "authors": [
    {
      "name": "Shi Feng"
    },
    {
      "name": "Gilles Mordant"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/exjq4xv66bmtp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26597",
  "fulltext_url": null,
  "id": "exjq4xv66bmtp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26597",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214088492",
  "pdf_url": "https://arxiv.org/pdf/2609.26597",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/exjq4xv66bmtp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26597",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26597"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Fluctuations of the quadratic matching cost on the flat torus: dimensions two, three and four",
  "title_text": "Fluctuations of the quadratic matching cost on the flat torus: dimensions two, three and four",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/exjq4xv66bmtp",
  "venue": "arXiv",
  "versions": []
}