Abstract
We match independent uniform samples from the squares [0, 1]2 and [2, 3] × [0, 1] by minimizing total Euclidean distance. The transport problem between the two uniform distributions has infinitely many optimal plans. We prove that, as the sample size grows, the empirical optimal plans converge weakly in probability to a single deterministic plan. This plan preserves height and has the same joint distribution of paired horizontal coordinates at every height. We prove convergence by establishing uniqueness of the stationary local matching obtained after rescaling and identifying the selected horizontal coupling from its source-Palm distribution.