Asymptotics for Random Quadratic Transportation Costs

TL;DR

This paper proves asymptotic limits for the quadratic transportation problem between random i.i.d. points in dimensions greater than three, extending previous results beyond two dimensions and uniform distributions.

Contribution

It introduces a new quantitative upper bound for quadratic optimal transportation, applicable to general random measures and higher dimensions, advancing the understanding of transportation costs.

Findings

Established asymptotic limits in dimensions >3

Developed a quantitative upper bound for transportation problems

Applicable to general random measures and Brownian paths

Abstract

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.