Dependent percolation in two dimensions

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) ∈ ℤ2 closed 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 (i0, j0) 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.