Point-shifts of Point Processes on Topological Groups

TL;DR

This paper studies flow-adapted point-shifts on topological groups, generalizing known Euclidean results to unimodular groups, and explores their properties, classifications, and connections to network theory.

Contribution

It extends the classification of connected components of point-shifts to unimodular groups and introduces isomodularity as a key concept for invariance and distributional properties.

Findings

Cardinality classification generalized to unimodular groups

Counterexample provided on a non-unimodular group

Connections established between point-shifts and vertex-shifts in networks

Abstract

This paper focuses on flow-adapted point-shifts of point processes on topological groups, which map points of a point process to other points of the point process in a translation invariant way. Foliations and connected components generated by point-shifts are studied, and the cardinality classification of connected components, previously known on Euclidean space, is generalized to unimodular groups. An explicit counterexample is also given on a non-unimodular group. Isomodularity of a point-shift is defined and identified as a key component in generalizations of Mecke's invariance theorem in the unimodular and non-unimodular cases. Isomodularity is also the deciding factor of when the reciprocal and reverse of a point-map corresponding to a bijective point-shift are equal in distribution. Next, sufficient conditions for separating points of a point process are given. Finally, connections between point-shifts of point processes and vertex-shifts of unimodular networks are given that allude to a deeper connection between the theories.