There is no stationary cyclically monotone Poisson matching in 2d

TL;DR

This paper proves that in two-dimensional space, there is no stationary cyclically monotone matching between two independent Poisson processes, using harmonic approximation and martingale techniques.

Contribution

It establishes a non-existence result for stationary cyclically monotone Poisson matchings in 2D, providing a new proof with martingale methods.

Findings

No stationary cyclically monotone matching exists in 2D for independent Poisson processes.

The proof combines harmonic approximation with local asymptotics.

A new self-contained proof using martingale arguments is provided.

Abstract

We show that there is no cyclically monotone stationary matching of two independent Poisson processes in dimension $d=2$. The proof combines the harmonic approximation result from \cite{GHO} with local asymptotics for the two-dimensional matching problem for which we give a new self-contained proof using martingale arguments.