Algorithms for general hard-constraint point processes via discretization

We study a general model for continuous spin systems with hard-core interactions. Our model allows for a mixture of q types of particles on a d-dimensional Euclidean region V of volume ν (V). For each type, particle positions are distributed according to a Poisson point process. 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 distance parameterized by a q×q matrix. This general model encompasses classical continuous spin systems, such as the hard-sphere model or the Widom–Rowlinson model. We present su cient conditions for approximating the partition function of this model, which is the normalizing factor of its Gibbs distribution. For the hard-sphere model, our method yields a randomized approximation algorithm with running time polynomial in ν (V) for the known uniqueness regime of the Gibbs measure. In the same parameter regime, we obtain a quasi-polynomial deterministic approximation algorithm for the hard-sphere model, which, to our knowledge, is the rst rigorous deterministic approximation algorithm for a continuous spin system. We obtain similar approximation results for all continuous spin systems captured by our model and, in particular, the rst explicit approximation bounds for the Widom–Rowlinson model. Additionally, we show how to obtain e cient approximate samplers for the Gibbs distributions of the respective spin systems within the same parameter regimes. Key to our method is reducing the continuous model to a discrete instance of the hard-core model with size polynomial in ν (V). This generalizes existing discretization schemes for the hard-sphere model and, additionally, improves the required number of vertices of the generated graph from super-exponential to quadratic in ν (V), which we argue to be tight.