From Combinatorics to Partial Differential Equations

TL;DR

This work analyzes the optimal matching problem for random point clouds in $\mathbb{R}^d$, focusing on the cost of optimally pairing two independent empirical measures drawn from a uniform distribution on a box. It connects combinatorial matching to optimal transport and PDE perspectives, revealing a sharp dimension-dependent scaling with a critical threshold at $d=2$. The authors establish tight upper and lower bounds for the asymptotic matching cost: the upper bound is obtained by constructing a transport map via Kantorovich relaxation and a dyadic, multi-scale decomposition, while the lower bound leverages the dual formulation and a carefully built dual potential through a dyadic inductive scheme. Together these results illuminate how the cost scales with the particle density and dimension, and they underscore the PDE/OT mechanisms underlying random geometric matching. The methods bridge probabilistic concentration, transport inequalities, and multi-scale constructions to yield precise, dimension-sensitive asymptotics.

Abstract

The optimal matching of point clouds in $\mathbb{R}^d$ is a combinatorial problem; applications in statistics motivate to consider random point clouds, like the Poisson point process. There is a crucial dependance on dimension $d$, with $d=2$ being the critical dimension. This is revealed by adopting an analytical perspective, connecting e.\,g.~to Optimal Transportation. These short notes provide an introduction to the subject. The material presented here is based on a series of lectures held at the International Max Planck Research School during the summer semester 2022. Recordings of the lectures are available at https://www.mis.mpg.de/events/event/imprs-ringvorlesung-summer-semester-2022.

Figures (10)

• Figure 1: The map $T$.
• Figure 2: A possible sequence $\{n_i,m_j\}$.
• Figure 3: Graph of $\rho_-$ and $T_{\rho_-}$.
• Figure 4: Dyadic decomposition.
• Figure 5: Density $\rho_k$.

Theorems & Definitions (11)

• Theorem 1.1
• Lemma 1.2
• proof
• Proposition 2.1
• proof : Proof of the upper bound of Theorem \ref{['thm:AKTasymptotic']}
• Lemma 2.2
• proof
• proof : Proof of Proposition \ref{['prop:mapupper']}
• Proposition 3.1
• proof : Proof of the lower bound of Theorem \ref{['thm:AKTasymptotic']}