Non-existence of the 2D quadratic Poisson optimal matching

This is my MSc thesis, where I analysed Otto, Huesmann and Mattesini’s paper showing the non-existence of stationary, non-ergodic, quadratic and locally-optimal 2D Poisson matchings. The proof is based on the fact that the local optimiality of the Poisson matching problem is equivalent to the discrete Monge-Kantorovich optimal transport problem between the restricted Poisson process counting measures. Analysing this problem and performing a multiscale argument, one arrives at a contradiction, showing that such a matching does not exist.

There is also a generalisation of this result to $\ell_p$ optimal matchings, where the cost is the $\ell_p$ norm; this is given in a follow-up paper by the same authors.

I also briefly discuss some open problems and possible extensions of this work, such as extending the result to multi-matchings (where the matching defines a hypergraph) and to multiple point processes in 2D.

Download my thesis here