TITLE: A percolation problem
QUESTION [26 upvotes]: Let's consider the 2-dimensional integer lattice $\mathbb{Z}^2$ for simplicity. In "ordinary" bond percolation, there is a parameter $p \in [0,1]$, and each edge is on with probability $p$. Consider now the following model : all edges are present, but each is to be given an orientation either away from or toward the origin (this is well-defined in $\mathbb{Z}^2$). Hence, each edge is oriented away from $(0,0)$ with probability $p$, and toward $(0,0)$ with probability $1-p$. The basic question is whether we percolate when $p > 1/2$, i.e.: if $p > 1/2$, is there a non-zero probability of there being an oriented path from $(0,0)$ to infinity ? The intuition behind this is that a biased 2-D random walk is known to be transient. I have seen other models of "oriented percolation" but not this one, so my question is really whether there are any results on this model ? By the way, it is known we don't percolate at $p = 1/2$. Here the model coincides with one that appears for example in several of Grimmett's books, where the bias is in a certain direction (north and east), rather than away from $(0,0)$. There it is also apparently open whether one percolates for $p > 1/2$.

REPLY [2 votes]: Not sure if that is what Linusson does, but it is not difficult to see that the cluster distribution at the origin at $p=1/2$ is exactly the same as for ordinary percolation at $p=1/2$ (in fact, that holds in any graph, in particular in $\mathbb Z^3$ there is percolation wpp at $p=1/2$; there is no use of duality).
Here is a coupling argument. Start with standard percolation, and explore the cluster of the origin by looking at every edge you need one by one. On the directed side of the coupling, if the edge is open, put it in the orientation that allows you to proceed, and if it is closed, put it in the reverse orientation. That builds an orientation on a random set of bonds, which you can complete into a configuration in the whole plane.
In particular, the average size of the cluster at $p=1/2$ is infinite. Maybe there is even Russo-Seymour-Welsh and so on, anyway it looks very critical. And there is "too much symmetry" in that for every $x$ in the plane, the cluster of $x$ has the same distribution in both models.
But indeed without monotonicity it is difficult to see how to proceed ...