{
  "abstract": "For the empirical measure of the first $n$ points of one infinite sequence of independent uniform samples in the unit square, we identify the leading constant in the bounded-Lipschitz distance to area measure, equivalently in $W_1$, both in expectation and almost surely along all sample sizes. The constant is $c_*=κ_1/(2\\sqrtπ)\\in(0,1/4]$, where $κ_1$ is the linear-growth Gaussian variational constant of Armegioiu--Goldman--Grotto--Trevisan. This answers the two-dimensional bounded-Lipschitz convergence question posed by Yukich in 1989. We give the complete endpoint argument, square--torus comparison, empirical reduction, and full-sequence probability argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21070",
  "author_total": 4,
  "authors": [
    {
      "name": "Weitao Huang"
    },
    {
      "name": "Renzhen Wang"
    },
    {
      "name": "Yichen Wu"
    },
    {
      "name": "Deyu Meng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b7nmmt6zmepyk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21070",
  "fulltext_url": null,
  "id": "b7nmmt6zmepyk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21070",
  "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": "W7222431009",
  "pdf_url": "https://arxiv.org/pdf/2610.21070",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b7nmmt6zmepyk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.21070",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21070"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions",
  "title_text": "Exact Endpoint Asymptotics and Almost-Sure Limits for Uniform Empirical Measures in Two Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/b7nmmt6zmepyk",
  "venue": "arXiv",
  "versions": []
}