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.

A note on partial rejection sampling for the hard disks model in the plane

“…Our analysis closely resembles that given in [3], with an extra ingredient. A theorem of Bollobas [2] states that if one pushes apart a collection of disks in R 2 (of equal radii) in a continuous fashion such that the pairwise distances between their centers are always increasing, then the area of their union is also increasing.…”

“…We use the same notation as [3]. Recall that P t is the set of points sampled during round t ≥ 0 of the algorithm.…”

“…It is shown in [3] that the hard disks process can be coupled to a Poisson process in such a way that the latter configuration always contains the former. In particular,…”

“…The authors of [3] conjecture that λ can be taken to be ≈ 0.5, according to their simulations. In this note, we slightly improve the value of λ (although it remains quite far from the conjectured truth):…”

“…In a very recent work of Guo and Jerrum [3], the general-purpose method of partial rejection sampling is applied to the problem of sampling from the hard disks model. The authors show that for sufficiently small λ > 0, this algorithm is efficient -namely it runs in expected O(r −2 ) time.…”