Perfect simulation of the hard disks model by partial rejection sampling

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.