Dependent percolation in two dimensions
Let \(k\) be a natural number and define an oriented site percolation on \(Z^2\) as follows. Let \(x_i\), \(y_j\) be independent random variables with values uniformly distributed in \(\{1,\ldots,k\}\). Declare a site \((i,j)\in Z^2\) closed if \(x_i=y_j\), and open otherwise. P. Winkler conjectured that if \(k\geq 4\), then with positive probability there is an infinite path \(P=(i_0,j_0)(i_1,j_1)\ldots\) such that \(0=i_0\leq i_1\leq \ldots\), \(0=j_0\leq j_1\leq \ldots\), and each site \((i_n,j_n)\) is open. This conjecture is still open. In this paper the authors prove the weaker result that if \(k\geq 4\), then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive.
scientific article; zbMATH DE number 1525164