随机欧几里德匹配问题渐近代价的收敛性
Convergence of asymptotic costs for random Euclidean matching problems
摘要：
研究了d $\ge$ 3维空间中均匀分布在单位立方体上的n个独立随机点的两个样本之间的二部匹配的平均最小代价，其中两点之间的匹配代价由它们的欧氏距离的任意幂p $\ge$ 1给出.随着n的增长，我们证明收敛，经过适当的重整化，朝着一个有限的正常数。我们还考虑了n点之间的最优输运和一致测度的类似问题。的证明相结合的次加性不等式与PDE ananomaly类似的背景下提出的匹配问题，在两个维度，后来扩展到更高的维度，以获得上界。
英文摘要：
We investigate the average minimum cost of a bipartite matching between two samples of n independent random points uniformly distributed on a unit cube in d $\ge$ 3 dimensions, where the matching cost between two points is given by any power p $\ge$ 1 of their Euclidean distance. As n grows, we prove convergence, after a suitable renormalization, towards a finite and positive constant. We also consider the analogous problem of optimal transport between n points and the uniform measure. The proofs combine sub-additivity inequalities with a PDE ansatz similar to the one proposed in the context of the matching problem in two dimensions and later extended to obtain upper bounds in higher dimensions.
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