Dependent percolation in two dimensions

For a natural number k, define an oriented site percolation on Z(2) as foll ows. Let x(i), y(j) be independent random variables with values uniformly d istributed in {1,..., k}. Declare a site (i, j) is an element of Z(2) close d if x(i) = y(j), and open otherwise. Peter Winkler conjectured some years ago that if k greater than or equal to 4 then with positive probability the re is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i(o), j(o))(i(1), j(1)) ... such that 0 = i(o) less than or equal to i(1) less than or equal to ..., 0 = j(o) less than or equal to j(1) less than or equal to ..., and each site (i(n),j(n)) 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 assertio n holds in the unoriented case: if k greater than or equal to 4 then the pr obability that there is an infinite path that starts at the origin and cons ists only of open sites is positive. Furthermore, we shall show that our me thod can be applied to a wide variety of distributions of (x(i)) and (y(j)) Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods.