Percolation problems appear in a large variety of different contexts ranging from the design of composite materials to vaccination strategies on community networks. The key observable for many applications is the percolation threshold. Unlike the universal critical exponents, the percolation threshold depends explicitly on the specific system properties. As a consequence, theoretical approaches to the percolation threshold are rare and generally tailored to the specific application. Yet, any percolating cluster forms a discrete network the emergence of which can be cast as a graph problem and analyzed using branching processes. We propose a general mapping of any kind of percolation problem onto a branching process which provides rigorous lower bounds of the percolation threshold. These bounds progressively tighten as we incorporate more information into the theory. We showcase our approach for different continuum problems finding accurate predictions with almost no effort. Our approach is based on first principles and does not require fitting parameters. As such it offers an important theoretical reference in a field that is dominated by simulation studies and heuristic fit functions.

中文翻译：

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关键渗透的通用方法

渗透问题出现在各种不同的环境中，从复合材料的设计到社区网络的疫苗接种策略。对于许多应用来说，可观察到的关键是渗透阈值。与通用临界指数不同，渗透阈值明确取决于特定的系统属性。因此，渗滤阈值的理论方法很少，并且通常针对特定应用进行定制。然而，任何渗透集群都会形成一个离散网络，其出现可以被视为图形问题并使用分支过程进行分析。我们提出了任何类型的渗滤问题到分支过程的一般映射，该分支过程提供了渗滤阈值的严格下限。随着我们将更多信息纳入理论中，这些界限逐渐收紧。我们展示了我们针对不同连续体问题的方法，几乎​​不费吹灰之力就能找到准确的预测。我们的方法基于第一原理，不需要拟合参数。因此，它为以模拟研究和启发式拟合函数为主的领域提供了重要的理论参考。