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 $\mathbb{V}$ of volume $\nu(\mathbb{V})$. For each type, particle positions are distributed according to a Poisson point process. The Gibbs distribution over possible system states is given by the mixture of these point processes conditioned that no two particles are closer than some distance parameterized by a $q \times q$ matrix. This model encompasses classical continuous spin systems, such as the hard-sphere model or the Widom-Rowlinson model. We present sufficient conditions for approximating the partition function of this model, which is the normalizing factor of its Gibbs measure. For the hard-sphere model, our method yields a randomized approximation algorithm with running time polynomial in $\nu(\mathbb{V})$ for the known uniqueness regime of the Gibbs measure. In the same parameter regime, we obtain a quasi-polynomial deterministic approximation, which, to our knowledge, is the first rigorous deterministic algorithm for a continuous spin system. We obtain similar approximation results for all continuous spin systems captured by our model and, in particular, the first explicit approximation bounds for the Widom-Rowlinson model. Additionally, we show how to obtain efficient 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 $\nu(\mathbb{V})$. 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 $\nu(\mathbb{V})$, which we argue to be tight.

翻译：我们研究的是具有硬核心互动作用的连续旋转系统的一般模型。 我们的模型允许在 $d$ 的 Euclidean 区域 $\\ mathb{V} 中混合 $nu (\\ mathb{V}) 体积。 对于每种类型, 粒子位置会按照 Poisson 点进程分布。 Gibs 分布于可能的系统状态, 这些点进程的混合条件是, 没有任何两个粒子会比由 $\ 时间 的离线参数更近。 这个模型包含传统的连续旋转系统, 如硬镜模型或Widom- Rowlinson 模型。 我们为这个模型提供了足够的条件, 我们的直径直径模型和连续的硬基数分析系统, 我们的直基数分析系统会通过一个连续的硬基数的硬基数, 不断的硬基数的基数的基数的基数, 我们的基数的基数的基数分析系统会得到一个连续的硬基数的基数, 我们的基数的基数的基数的基数的基数的基数, 我们的基数的基数的基数的基数的基数的基数的基数的基数系统会是不断的基数的基数的基数, 我们的基数的基数的基数的基数的基数的基数的基数的基数的基数的基数的基数的基数的基数, 。