We consider the two-dimensional random matching problem in $${\mathbb {R}}^2.$$ In a challenging paper, Caracciolo et al. Phys Rev E 90(1):012118 (2014), on the basis of a subtle linearization of the Monge-Ampere equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent 2, between two samples of N uniformly distributed points in the unit square is $$\log N/2\pi N$$ plus corrections, while the expected value of the square of the Wasserstein distance between one sample of N uniformly distributed points and the uniform measure on the square is $$\log N/4\pi N$$ . These conjectures have been proved by Ambrosio et al. Probab Theory Rel Fields 173(1–2):433–477 (2019). Here we consider the case in which the points are sampled from a non-uniform density. For first we give formal arguments leading to the conjecture that if the density is regular and positive in a regular, bounded and connected domain $$\Lambda $$ in the plane, then the leading term of the expected values of the Wasserstein distances are exactly the same as in the case of uniform density, but for the multiplicative factor equal to the measure of $$\Lambda $$ . We do not prove these results but, in the case in which the domain is a square, we prove estimates from above that coincides with the conjectured result.

中文翻译：

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非恒定密度的 2D 欧几里德随机匹配

我们考虑了 $${\mathbb {R}}^2.$$ 中的二维随机匹配问题。在一篇具有挑战性的论文中，Caracciolo 等人。Phys Rev E 90(1):012118 (2014)，基于 Monge-Ampere 方程的微妙线性化，推测两个 N 的样本之间的 Wasserstein 距离平方的期望值，指数为 2单位正方形中的分布点为$$\log N/2\pi N$$加上修正，而N个均匀分布点的一个样本与正方形上的均匀测度之间的Wasserstein距离的平方的期望值为$ $\log N/4\pi N$$ 。这些猜想已被 Ambrosio 等人证明。概率论 Rel Fields 173(1–2):433–477 (2019)。这里我们考虑从非均匀密度采样点的情况。首先，我们给出了导致以下猜想的形式论据：如果密度在平面中的正则、有界和连通域 $$\Lambda $$ 中是正则正的，那么 Wasserstein 距离的期望值的前导项正好是与均匀密度的情况相同，但乘法因子等于 $$\Lambda $$ 的度量。我们不证明这些结果，但在域是正方形的情况下，我们证明上面的估计与推测结果一致。