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.
Highlights
The hard disks model is one of the simplest gas models in statistical physics
In this paper, we take this region to be the unit square [0, 1]2. This model was precisely the one studied by Metropolis et al [12], in their pioneering work on the Markov chain Monte Carlo (MCMC) method
69:2 Perfect Simulation of the Hard Disks Model by Partial Rejection Sampling according to a Poisson point process of intensity λr = λ/(πr2), conditioned on the disks being non-overlapping
Summary
The hard disks model is one of the simplest gas models in statistical physics. Its configurations are non-overlapping disks of uniform radius r in a bounded region of R2. 69:2 Perfect Simulation of the Hard Disks Model by Partial Rejection Sampling according to a Poisson point process of intensity λr = λ/(πr2), conditioned on the disks being non-overlapping. The challenge is the following: produce a realisation P ⊂ [0, 1]2 of a Poisson point process of intensity λr in the unit square, conditioned on the event that no pair of points in P are closer than 2r in Euclidean distance We refer to this target measure as the hard disks distribution. The only rigorous guarantee of this approach, via [6], is α(0.21027) > 0.0887
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