Algorithms for Hard-Constraint Point Processes via Discretization

Abstract

We study the algorithmic applications of a natural discretization for the hard-sphere model and the Widom–Rowlinson model in a region of $$d $$ -dimensional Euclidean space $$ \mathbb {V} \subset \mathbb {R} ^{d}$$ . These continuous models are frequently used in statistical physics to describe mixtures of one or multiple particle types subjected to hard-core interactions. For each type, particles are distributed according to a Poisson point process with a type-specific activity parameter, called fugacity. The Gibbs distribution over all possible system states is characterized by the mixture of these point processes conditioned that no two particles are closer than some type-dependent distance threshold. A key part in better understanding the Gibbs distribution is its normalizing constant, called partition function. Our main algorithmic result is the first deterministic approximation algorithm for the partition function of the hard-sphere model and the Widom–Rowlinson model in box-shaped regions of Euclidean space. Our algorithms have quasi-polynomial running time in the volume of the region $$\nu \left( \mathbb {V} \right) $$ if the fugacity is below a certain threshold. For the $$d $$ -dimensional hard-sphere model with particles of unit volume, this threshold is $$\textrm{e}/2^d $$ . As the number of dimensions $$d $$ increases, this bound asymptotically matches the best known results for randomized approximation of the hard-sphere partition function. We prove similar bounds for the Widom–Rowlinson model. To the best of our knowledge, this is the first rigorous algorithmic result for this model.