论文标题
随机连接模型中的直接连接函数
The Direct-Connectedness Function in the Random Connection Model
论文作者
论文摘要
我们研究了强度幂的连续渗透的随机连接模型中连接性函数的扩展。确切地说，我们研究了配对连接性和直接连接性函数，这些函数通过ornstein-zernike方程相互关联。我们展示了一个事实，即扩展的系数由连接的和2美元连接的图形组成。在物理学文献中，众所周知，这种情况是基于吉布斯点过程的渗透模型，并且与液态统计力学中相关功能开发的形式主义类似。 我们找到了直接连接函数的表示，并在强度上的边界使我们能够传递到热力学极限。在某些情况下（例如，在高维度）中，结果几乎在整个亚临界体制中都是有效的。此外，我们将这些扩展与物理学文献联系起来，并展示了它们与蕾丝膨胀所提供的表达方式。
We investigate expansions for connectedness functions in the random connection model of continuum percolation in powers of the intensity. Precisely, we study the pair-connectedness and the direct-connectedness functions, related to each other via the Ornstein-Zernike equation. We exhibit the fact that the coefficients of the expansions consist of sums over connected and $2$-connected graphs. In the physics literature, this is known to be the case more generally for percolation models based on Gibbs point processes and stands in analogy to the formalism developed for correlation functions in liquid-state statistical mechanics. We find a representation of the direct-connectedness function and bounds on the intensity which allow us to pass to the thermodynamic limit. In some cases (e.g., in high dimensions), the results are valid in almost the entire subcritical regime. Moreover, we relate these expansions to the physics literature and we show how they coincide with the expression provided by the lace expansion.
