We establish the validity of asymptotic limits for the general transportation problem between random i.i.d. points and their common distribution, with respect to the squared Euclidean distance cost, in any dimension larger than three. Previous results were essentially limited to the two (or one) dimensional case, or to distributions whose absolutely continuous part is uniform. The proof relies upon recent advances in the stability theory of optimal transportation, combined with functional analytic techniques and some ideas from quantitative stochastic homogenization. The key tool we develop is a quantitative upper bound for the usual quadratic optimal transportation problem in terms of its boundary variant, where points can be freely transported along the boundary. The methods we use are applicable to more general random measures, including occupation measure of Brownian paths, and may open the door to further progress on challenging problems at the interface of analysis, probability, and discrete mathematics.

翻译：本文确立了在任意大于三维的空间中，对于服从独立同分布的随机点与其共同分布之间的一般运输问题，关于平方欧几里得距离成本，其渐近极限的有效性。先前的研究结果主要局限于二维（或一维）情形，或局限于其绝对连续部分为均匀分布的分布。证明依赖于最优运输稳定性理论的最新进展，结合了泛函分析技术以及定量随机均匀化中的一些思想。我们开发的关键工具是针对通常二次最优运输问题的一个定量上界，该上界以其边界变体（其中点可以沿边界自由运输）表示。我们所使用的方法适用于更一般的随机测度，包括布朗路径的占有测度，并可能为分析、概率论和离散数学交叉领域的挑战性问题开启进一步进展之门。