标题： 渗流和刚度渗流中的自相似间隙动力学 显示英文标题

标题： Self-similar gap dynamics in percolation and rigidity percolation

摘要： 空间自相似性是临界现象的特征。 我们研究渗流的动力过程，在此过程中，逐步将键插入到空格点上，直到完全占据，同时跟踪描述聚类大小变化的间隙。 令人惊讶的是，我们发现间隙大小在整个过程或其显著部分内遵循普遍的幂律分布，揭示了一种之前未被认识到的时间自相似性。 这种现象出现在各种渗流模型中，如标准渗流、爆炸性渗流和刚度渗流。 此外，在刚度渗流中，我们直接观察到由单个键插入触发的级联聚类合并动力学，并进一步在合并聚类的数量中获得了独特的时间自相似性，这些在静态分析中是隐藏的。 我们的结果还表明，对于刚度渗流，时间自相似性可能比空间自相似性更内在。 这些发现为临界现象提供了新的视角，并拓宽了在复杂系统中的潜在应用。 显示英文摘要

摘要： Spatial self-similarity is a hallmark of critical phenomena. We investigate the dynamic process of percolation, in which bonds are incrementally inserted to an empty lattice until fully occupied, and track the gaps describing the changes in cluster sizes. Surprisingly, we find that the gap sizes follow a universal power-law distribution throughout the whole or a significant portion of process, revealing a previously unrecognized temporal self-similarity. This phenomenon appears across various percolation models, like standard, explosive and rigidity percolation. Furthermore, in rigidity percolation, we directly observe a cascading cluster-merging dynamics, triggered by single bond insertion, and further obtain a distinct temporal self-similarity in the number of merged clusters, which are hidden in static analyses. Our results also suggest that, for rigidity percolation, the temporal self-similarity is probably more intrinsic than the spatial one. These findings offer a fresh perspective on critical phenomena and broaden potential applications across complex systems.