论文标题
具有一维远程相互作用的超顺从晶格上单词的渗透
Percolation of words on the hypercubic lattice with one-dimensional long-range interactions
论文作者
论文摘要
我们研究了随机环境中单词渗透的问题。对于每个顶点，我们独立地分配了一个字母$ 0 $或$ 1 $，根据bernoulli r.v.的参数$ p $。环境是从$ \ Mathbb {z}^{d-1} \ times \ times \ Mathbb {z} $，$ d \ geq 3 $上获得的独立远程债券渗透配置获得的结果图，其中每个边缘平行于$ \ mathbb {z}^{d-d-1} $ pob y是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n是$ n ins pob n is pob n ins pob n ins pob n ins pob lenge pobs $ \ mathbb {z} $开放，概率$ p_n $。我们证明，如果$ p_n $的总和差异，那么对于任何$ε$和$ p $，即使所有具有大于$ k $的连接的连接都被抑制了所有单词，因此从原点看到所有单词。
We investigate the problem of percolation of words in a random environment. To each vertex, we independently assign a letter $0$ or $1$ according to Bernoulli r.v.'s with parameter $p$. The environment is the resulting graph obtained from an independent long-range bond percolation configuration on $\mathbb{Z}^{d-1} \times \mathbb{Z}$, $d\geq 3$, where each edge parallel to $\mathbb{Z}^{d-1}$ has length one and is open with probability $ε$, while edges of length $n$ parallel to $\mathbb{Z}$ are open with probability $p_n$. We prove that if the sum of $p_n$ diverges, then for any $ε$ and $p$, there is a $K$ such that all words are seen from the origin with probability close to $1$, even if all connections with length larger than $K$ are suppressed.
