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On the Asymptotic W1W_1 Cost of Two-Dimensional Semi-Discrete Matching

Primarily AI-generated textHuman understanding: no partsmath.PR — Probabilitymath.AP — Analysis of PDEs

Contributed by Heng Ma ↗

SubmitterHeng Ma

Version 1 / Sep 30, 2026 / CC BY 4.0

Abstract

Let X1,X2,…X_1,X_2,\ldots be independent uniform points in the unit square and let λ\lambda denote Lebesgue probability measure. We prove that the normalized expected W1W_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 UU is unique under certain growth and regularity conditions, and the limiting constant c⋆=U(1,0)/2c_\star=U(1,0)/\sqrt2.

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References

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Version history

  1. v1Initial depositCurrentSep 30, 2026