Dependent percolation in two dimensions
Dependent percolation in two dimensions
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DOI：
10.1007/pl00008732
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发表时间：
2000
影响因子：
2
通讯作者：
A. Stacey
中科院分区：
文献类型：
--
作者：
P. Balister;B. Bollobás;A. Stacey
抽象的。对于自然数 k，定义 ℤ2 上的定向位点渗透如下。令 xi, yj 为独立随机变量，其值均匀分布在 {1, …, k} 中。如果 xi = yj 则声明一个站点 (i, j) εℤ2 闭合，否则打开。 Peter Winkler 几年前推测，如果 k≥ 4，则以正概率存在一条从原点开始的无限定向路径，所有站点都是开放的。即，存在一条无限路径 P = (i0, j0)(i1, j1) · · ·，使得 0 = i0≤i1≤· · ·，0 = j0≤j1≤· · ·，并且每个站点 (in, jn) 都是开放的。令人惊讶的是，这个猜想仍然是开放的：事实上，我们不知道这个猜想对于任何 k 值是否成立。在本文中，我们将证明相应断言在无向情况下成立的较弱结果：如果 k≤ 4，则存在从原点开始且仅由开放站点组成的无限路径的概率为正。此外，我们将证明我们的方法可以应用于 (xi) 和 (yj) 的各种分布。 Peter Winkler [14] 最近通过不同的方法独立地证明了各种类似的断言。
Abstract. For a natural number k, define an oriented site percolation on ℤ2 as follows. Let xi, yj be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ2closed if xi = yj, and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i0, j0)(i1, j1) · · · such that 0 = i0≤i1≤· · ·, 0 = j0≤j1≤· · ·, and each site (in, jn) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (xi) and (yj). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods.