Compact orbit spaces in Hilbert spaces and limits of edge-colouring models

TL;DR

This paper establishes conditions under which orbit spaces in Hilbert spaces are compact and applies these results to prove the existence of limits in edge-colouring models, extending the theory of graph limits.

Contribution

It introduces a new compactness criterion for orbit spaces in Hilbert spaces and applies it to demonstrate the existence of limits in edge-colouring models, answering a question by Lovász.

Findings

Orbit space $W/G$ is compact under certain conditions.

Limits of edge-colouring models exist, extending graph limit theory.

Provides a bridge between vertex- and edge-colouring models.

Abstract

Let $G$ be a group of orthogonal transformations of a real Hilbert space $H$. Let $R$ and $W$ be bounded $G$-stable subsets of $H$. Let $\|.\|_R$ be the seminorm on $H$ defined by $\|x\|_R:=\sup_{r\in R}|\langle r,x\rangle|$ for $x\in H$. We show that if $W$ is weakly compact and the orbit space $R^k/G$ is compact for each $k\in\oN$, then the orbit space $W/G$ is compact when $W$ is equiped with the norm topology induced by $\|.\|_R$. As a consequence we derive the existence of limits of edge-colouring models which answers a question posed by Lov\'asz. It forms the edge-colouring counterpart of the graph limits of Lov\'asz and Szegedy, which can be seen as limits of vertex-colouring models. In the terminology of de la Harpe and Jones, vertex- and edge-colouring models are called `spin models' and `vertex models' respectively.