论文标题
三维2临界引导渗透：稳定的套件方法
Three-dimensional 2-critical bootstrap percolation: The stable sets approach
论文作者
论文摘要
考虑一个$ p $ -random子集$ a $最初在离散的立方体$ [l]^3 $中最初感染的顶点，并假设每个顶点的附近由$ a_i $ the $ \ pm e_i $ tirections中的$ a_i $ the $ i \ in \ in \ in \ in \ in \ in \ in \ {1,2,3 \ \ a_______3 $ a_ 3 $ a a a a a a a a a a___1 \ a_ 3 \ a_ 3 \ a_ 3 $ a a a a a a a a a a a a a a_1假设我们在[l]^3 $中感染了任何健康的顶点$ v \已经感染了$ r $感染的邻居，并且感染的地点仍然永远被感染。在本文中，我们确定渗透率的关键长度的$ \ log $，最高为恒定因素，对于所有$ r \ in \ {a_3+1，\ dots，a_3+a_2 \} $，带有$ a_3 \ ge a_1+a_1+a_2 $。此外，我们在所有剩余情况下给出了上限$ a_3 <a_1+a_2 $，并相信它们紧紧地达到了一个恒定的因素。
Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^3$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,3\}$, where $a_1\le a_2\le a_3$. Suppose we infect any healthy vertex $v\in [L]^3$ already having $r$ infected neighbours, and that infected sites remain infected forever. In this paper we determine $\log$ of the critical length for percolation up to a constant factor, for all $r\in \{a_3+1, \dots, a_3+a_2\}$ with $a_3\ge a_1+a_2$. We moreover give upper bounds for all remaining cases $a_3 < a_1+a_2$ and believe that they are tight up to a constant factor.
