Using random graphs to sample repulsive Gibbs point processes with arbitrary-range potentials

TL;DR

This paper introduces a novel reduction method that transforms repulsive Gibbs point processes into hard-core models on random geometric graphs, enabling efficient sampling and partition function approximation for arbitrary-range potentials.

Contribution

It presents the first approach to handle pair potentials of unbounded range by reducing Gibbs point processes to the well-studied hard-core model on random geometric graphs.

Findings

Partition function concentrates around the hard-core model's partition function.

Algorithms for the hard-core model can be adapted for Gibbs point processes.

The approach extends the applicability of existing algorithms to unbounded-range potentials.

Abstract

We study computational aspects of repulsive Gibbs point processes, which are probabilistic models of interacting particles in a finite-volume region of space. We introduce an approach for reducing a Gibbs point process to the hard-core model, a well-studied discrete spin system. Given an instance of such a point process, our reduction generates a random graph drawn from a natural geometric model. We show that the partition function of a hard-core model on graphs generated by the geometric model concentrates around the partition function of the Gibbs point process. Our reduction allows us to use a broad range of algorithms developed for the hard-core model to sample from the Gibbs point process and approximate its partition function. This is, to the extend of our knowledge, the first approach that deals with pair potentials of unbounded range. We compare the resulting algorithms with recently established results and study further properties of the random geometric graphs with respect to the hard-core model.