Perfect simulation of the Hard Disks Model by Partial Rejection Sampling

TL;DR

This paper introduces a perfect simulation technique for the hard disks model using partial rejection sampling, achieving efficient exact sampling under certain density conditions and extending to higher dimensions.

Contribution

It develops a novel application of partial rejection sampling for perfect simulation in continuous models like hard disks and spheres, with a new analysis framework.

Findings

Produces exact samples in O(log n) rounds

Extends to hard spheres in higher dimensions

Efficient for moderate disk densities

Abstract

We present a perfect simulation of the hard disks model via the partial rejection sampling method. Provided the density of disks is not too high, the method produces exact samples in $O(\log n)$ rounds, and total time $O(n)$, where $n$ is the expected number of disks. The method extends easily to the hard spheres model in $d>2$ dimensions. In order to apply the partial rejection method to this continuous setting, we provide an alternative perspective of its correctness and run-time analysis that is valid for general state spaces.