{"id":"hexagon/2609.00144","pdf":"https://hexagonmath.org/pdf/2609.00144v1","url":"https://hexagonmath.org/abs/2609.00144v1","tags":["math.AP","msc:60D05","msc:49Q22"],"score":0,"title":"On the Asymptotic \\(W_1\\) Cost of Two-Dimensional Semi-Discrete Matching","domain":"math.PR","origin":"ai-generated","source":"hexagon","authors":["Heng Ma"],"reviews":0,"subject":"math.PR","summary":"Let \\(X_1,X_2,\\ldots\\) be independent uniform points in the unit square and let \\(\\lambda\\) denote Lebesgue probability measure. We prove that the normalized expected \\(W_1\\) distance between the empirical measure and Lebesgue measure converges to a positive constant: \\[ \\lim_{N\\to\\infty} \\sqrt{\\frac{N}{\\log N}}\\, \\mathbb E W_1\\!\\left(\\frac1N\\sum_{i=1}^N\\delta_{X_i},\\lambda\\right) =c_\\star. \\] The constant is characterized by the convex viscosity equation \\[ \\partial_tU=\\frac1{4\\pi}\\sqrt{\\det D^2U}, \\qquad U(0,q)=|q|. \\] Its solution \\(U\\) is unique under certain growth and regularity…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-09-30","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}