# Rejection and Importance Sampling based Perfect Simulation for Gibbs hard-sphere models

## TL;DR

This paper introduces an importance sampling rejection method for perfect simulation of Gibbs hard-sphere models, providing theoretical and numerical comparisons with existing algorithms.

## Contribution

It proposes a novel importance sampling based rejection approach for perfect sampling of Gibbs hard-sphere models, analyzing its efficiency and asymptotic complexity.

## Key findings

- The new method outperforms naive rejection algorithms in efficiency.
- Asymptotic analysis shows favorable complexity as sphere radius decreases.
- Numerical results confirm the theoretical advantages of the proposed approach.

## Abstract

Coupling from the past (CFTP) methods have been used to generate perfect samples from finite Gibbs hard-sphere models, an important class of spatial point processes, which is a set of spheres with the centers on a bounded region that are distributed as a homogeneous Poisson point process (PPP) conditioned that spheres do not overlap with each other. We propose an alternative importance sampling based rejection methodology for the perfect sampling of these models. We analyze the asymptotic expected running time complexity of the proposed method when the intensity of the reference PPP increases to infinity while the (expected) sphere radius decreases to zero at varying rates. We further compare the performance of the proposed method analytically and numerically with a naive rejection algorithm and popular dominated CFTP algorithms. Our analysis relies upon identifying large deviations decay rates of the non-overlapping probability of spheres whose centers are distributed as a homogeneous PPP.