Sharp PDE estimates for random two-dimensional bipartite matching with power cost function

TL;DR

This paper extends PDE-based analysis to the two-dimensional bipartite matching problem with convex power costs, linking PDE solutions to optimal transport costs and revealing new asymptotic behaviors.

Contribution

It generalizes the PDE approach from quadratic costs to convex power costs, introducing a non-linear q-Poisson equation for better analysis.

Findings

Established asymptotic relations between PDE energy and transport cost.

Extended PDE ansatz to non-linear q-Poisson equations.

Provided new insights into the asymptotic behavior of optimal matching costs.

Abstract

We investigate the random bipartite optimal matching problem on a flat torus in two-dimensions, considering general strictly convex power costs of the distance. We extend the successful ansatz first introduced by Caracciolo et al. for the quadratic case, involving a linear Poisson equation, to a non-linear equation of $q$-Poisson type, allowing for a more comprehensive analysis of the optimal transport cost. Our results establish new asymptotic connections between the energy of the solution to the PDE and the optimal transport cost, providing insights on their asymptotic behavior.