Gibbsian representation for point processes via hyperedge potentials

TL;DR

This paper develops a framework for representing marked point processes using hyperedge potentials, connecting stochastic geometry, measure transformations, and statistical mechanics.

Contribution

It introduces a method to construct absolutely-summable Hamiltonians for point processes via hyperedge potentials under locality conditions.

Findings

Successfully constructs hyperedge potential representations for point processes.

Links point process theory with statistical mechanics and measure transformations.

Provides potential representations for the Widom-Rowlinson model.

Abstract

We consider marked point processes on the d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing absolutely-summable Hamiltonians in terms of hyperedge potentials in the sense of Georgii et al. These potentials are a natural generalization of physical multi-body potentials which are useful in models of stochastic geometry. We prove that such representations can be achieved, under appropriate locality conditions of the specification. As an illustration we also provide such potential representations for the Widom-Rowlinson model under independent spin-flip time-evolution. Our paper draws a link between the abstract theory of point processes in infinite volume, the study of measures under transformations, and statistical mechanics of systems of point particles.