  We study algorithmic applications of a natural discretization for the
hard-sphere model and the Widom-Rowlinson model in a region
$\mathbb{V}\subset\mathbb{R}^d$. These models are used in statistical physics
to describe mixtures of one or multiple particle types subjected to hard-core
interactions. For each type, particles follow a Poisson point process with a
type specific activity parameter (fugacity). The Gibbs distribution is
characterized by the mixture of these point processes conditioned that no two
particles are closer than a type-dependent distance threshold. A key part in
better understanding the Gibbs distribution is its normalizing constant, called
partition function.
  We give sufficient conditions that the partition function of a discrete
hard-core model on a geometric graph based on a point set $X \subset
\mathbb{V}$ closely approximates those of such continuous models. Previously,
this was only shown for the hard-sphere model on cubic regions $\mathbb{V}=[0,
\ell)^d$ when $X$ is exponential in the volume of the region $\nu(\mathbb{V})$,
limiting algorithmic applications. In the same setting, our refined analysis
only requires a quadratic number of points, which we argue to be tight.
  We use our improved discretization results to approximate the partition
functions of the hard-sphere model and the Widom-Rowlinson efficiently in
$\nu(\mathbb{V})$. For the hard-sphere model, we obtain the first
quasi-polynomial deterministic approximation algorithm for the entire fugacity
regime for which, so far, only randomized approximations are known.
Furthermore, we simplify a recently introduced fully polynomial randomized
approximation algorithm. Similarly, we obtain the best known deterministic and
randomized approximation bounds for the Widom-Rowlinson model. Moreover, we
obtain approximate sampling algorithms for the respective spin systems within
the same fugacity regimes.
