菱形镶嵌上的自举渗流
Bootstrap percolation on rhombus tilings
AI总结：
本文将网格上2-自举渗流的几乎必然完全感染结果推广到任意平面菱形镶嵌图（含彭罗斯镶嵌等非周期情形），通过链凸性几何工具和统一概率界证明了更广测度类下的几乎必然感染。
摘要：
图上的2-自举渗流是一种扩散过程：当一个顶点至少有2个已感染邻居时，该顶点就会被感染，并且此后永远保持感染状态。从van Enter的开创性结果开始，人们已经在无限网格上针对随机伯努利初始配置对该过程进行了大量研究，该结果确立了对于任意非平凡的初始感染概率，整个网格几乎必然会被完全感染。在本文中，我们将这一结果推广到平面上任意菱形镶嵌的任意邻接图，包括诸如彭罗斯镶嵌这样的非周期镶嵌。实际上，我们证明了对于比非平凡伯努利测度更大的一类测度，整个图几乎必然被感染。我们的证明策略结合了基于链凸性的感染簇几何工具包，以及对特定几何模式的统一概率界，这些几何模式扮演了0-1律或遍历性的角色，而由于所考虑图缺乏对称性，这些工具在我们的设定中并不可用。
英文摘要：
2-boostrap percolation on a graph is a diffusion process where a vertex gets infected whenever it has at least 2 infected neighbours, and then stays infected forever. It has been much studied on the infinite grid for random Bernoulli initial configurations, starting from the seminal result of van Enter that establishes that the entire grid gets almost surely entirely infected for any non-trivial initial probability of infection. In this paper, we generalize this result to any adjacency graph of any rhombus tiling of the plane, including aperiodic ones like Penrose tilings. We actually show almost sure infection of the entire graph for a larger class of measure than non-trivial Bernoulli ones. Our proof strategy combines a geometry toolkit for infected clusters based on chain-convexity, and uniform probabilistic bounds on particular geometric patterns that play the role of 0-1 laws or ergodicity, which are not available in our settings due to the lack of symmetry of the graph considered.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
