吉尔伯特盘模型与地质统计学标记
Gilbert's disc model with geostatistical marking
AI总结：
本文研究了吉尔伯特盘模型中随机场对覆盖概率和渗流临界阈值的影响，发现空间依赖性可能增强或减弱渗流阈值。
摘要：
我们研究了一种吉尔伯特盘模型的变种，其中盘的位置由实数平面$\mathbb{R}^2$上的泊松过程确定，半径由一个底层的平稳和遍历随机场$φ:\mathbb{R}^2\to[0,\infty)$决定，该随机场与泊松过程独立。当随机场与点过程独立时，通常称为'地质统计学标记'。我们考察了在随机几何和渗流理论中感兴趣的典型性质，如覆盖概率和长程连接的存在性，在吉尔伯特模型中半径由某些随机场决定与半径独立但具有相同边缘分布的吉尔伯特模型之间有何不同。我们的主要观察之一是$\mathbb{R}^2$的完全覆盖不必然同时发生，且由随机场引起的空间依赖性可能既增加又减少渗流的临界阈值。
英文摘要：
We study a variant of Gilbert's disc model, in which discs are positioned at the points of a Poisson process in $\mathbb{R}^2$ with radii determined by an underlying stationary and ergodic random field $φ:\mathbb{R}^2\to[0,\infty)$, independent of the Poisson process. When the random field is independent of the point process one often talks about 'geostatistical marking'. We examine how typical properties of interest in stochastic geometry and percolation theory, such as coverage probabilities and the existence of long-range connections, differ between Gilbert's model with radii given by some random field and Gilbert's model with radii assigned independently, but with the same marginal distribution. Among our main observations we find that complete coverage of $\mathbb{R}^2$ does not necessarily happen simultaneously, and that the spatial dependence induced by the random field may both increase as well as decrease the critical threshold for percolation.
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