Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice
Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice
复制标题
几乎所有的词都出现在三角格子的关键位点渗透中
DOI：
10.1214/ejp.v3-32
复制
发表时间：
1998
影响因子：
1.4
通讯作者：
Yu Zhang
中科院分区：
文献类型：
--
作者：
H. Kesten;V. Sidoravicius;Yu Zhang
我们考虑三角形格上的临界点渗流，即对于三角形格的所有顶点v，我们分别以1/2的概率选择X（v）= 0或1。我们说，如果在满足$X（v_i）= \xi_i，i \ge 1$的三角格上存在自回避路径$（v_1，v_2，\dots）$，则在渗流构形中可以看到字$（\xi_1，\xi_2，\dots）\in \{0，1\}^{\Bbb N}$.我们证明，以概率1“几乎所有”的话，以及所有的周期性的话，除了两个字$（1，1，1，\dots）$和$（0，0，0，\dots）$，看到。这里的“几乎所有”是指关于度量$\mu_\beta$的几乎所有，在该度量下，$\xi_i$是i.i.d.。其中$\mu_\beta {\xi_i = 0}=1 - \mu_\beta {\xi_i = 1} = \beta$（对于任意的$0 <\beta < 1$）。
We consider critical site percolation on the triangular lattice, that is, we choose $X(v) = 0$ or 1 with probability 1/2 each, independently for all vertices $v$ of the triangular lattice. We say that a word $(\xi_1, \xi_2,\dots) \in \{0,1\}^{\Bbb N}$ is seen in the percolation configuration if there exists a selfavoiding path $(v_1, v_2, \dots)$ on the triangular lattice with $X(v_i) = \xi_i, i \ge 1$. We prove that with probability 1 "almost all" words, as well as all periodic words, except the two words $(1,1,1, \dots)$ and $(0,0,0,\dots)$, are seen. "Almost all" words here means almost all with respect to the measure $\mu_\beta$ under which the $\xi_i$ are i.i.d. with $\mu_\beta {\xi_i = 0}=1 - \mu_\beta {\xi_i = 1} = \beta$ (for an arbitrary $0 <\beta < 1$).