On the Asymptotic \(W_1\) Cost of Two-Dimensional Semi-Discrete Matching
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…
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