论文标题
$ d $二维引导性渗透模型，阈值至少双重指数
The $d$-dimensional bootstrap percolation models with threshold at least double exponential
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]^d $中最初受感染的顶点\ dots \ le a_d $。假设我们在[l]^d $已经感染了$ r $感染的邻居中感染了任何健康的顶点$ v \，并且感染的地点仍然被永远感染。在本文中，我们确定$（d-1）$ - 次数为渗透率的关键长度的对数达到不变的因素，对于所有$ d $ -tuples $（a_1，\ dots，a_d）$和所有$ r \ in \ {a_2+\ dots+a_2+\ dots+a_d+a_d+a_d+a_d+a_d+a_d+a_d+a_d+a_d+a_d+dots $ 此外，我们减少了确定所有$ d \ ge 3 $和所有$ r \ in \ {a_d+1，\ dots，a_1+a_1+a_2+dots+dots+a_d \} $的问题（粗大的）阈值，以确定所有$ d \ ge 3 $ and $ r \ r \ r \ in_________________________ + a_d \} $。
Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^d$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,\dots, d\}$, where $a_1\le a_2\le \dots \le a_d$. Suppose we infect any healthy vertex $v\in [L]^d$ already having $r$ infected neighbours, and that infected sites remain infected forever. In this paper we determine the $(d-1)$-times iterated logarithm of the critical length for percolation up to a constant factor, for all $d$-tuples $(a_1,\dots ,a_d)$ and all $r\in \{a_2+\dots + a_d+1, \dots, a_1+a_2+\dots + a_d\}$. Moreover, we reduce the problem of determining this (coarse) threshold for all $d\ge 3$ and all $r\in \{a_d+1, \dots, a_1+a_2+\dots + a_d\}$, to that of determining the threshold for all $d\ge 3$ and all $r\in \{ a_d+1, \dots, a_{d-1} + a_d\}$.
