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 $p > 1/2$.
[Math] A percolation problem
pr.probability
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James correctly identified percolation theory as the place where something like this is studied seriously. But let's do an elementary calculation.
Each possible path consists of $4n-1$ squares and is uniquely specified by saying which $2n-1$ of the $4n-2$ squares other than the first is vertically above the square before. Thus, there are exactly $$\binom{4n-2}{2n-1}$$ possible paths. Each path appears in a random board with probability $2^{-4n+1}$. Therefore, the expected number of paths is $$2^{-4n+1}\binom{4n-2}{2n-1} \sim \frac{1}{\sqrt{8\pi n}},$$ where the last expression comes from Stirling's formula.
Since the expected number of paths goes to 0, the probability that there is at least one path goes to 0 at least as fast. A quick simulation shows that James is correct that the probability goes to 0 exponentially fast (maybe slightly faster than $2^{-n}$).
I will first construct a simple deterministic example of a recurrent graph with $p_c^\mathrm{site}=0$ and then show how it can be modified to be a unimodular random rooted graph (see e.g. Nicolas Curien's lecture notes for the definition https://drive.google.com/file/d/16qEMGJU2g01g4YWkYtVKTSqetk-SpRmy/view)
Suppose we take a half-infinite line indexed by $\mathbb{N}=\{1,2,3,\ldots\}$ and, for each $n$, replace the vertex labelled $n$ with a complete graph of size $f(n)$ for some function $f:\mathbb{N}\to\mathbb{N}$ to be chosen later, where every vertex in the $n$th complete graph is connected to every vertex in the adjacent complete graphs at $n$ and $n-1$. In this example, site percolation occurs if there is at least one open vertex in the complete graph at $n$ for every sufficiently large $n$. From this and Borel-Cantelli, we see that $p_c^\mathrm{site}<1$ when $f(n)=\Omega(\log n)$ and $p_c^\mathrm{site}=0$ when $f(n)=\omega(\log n)$. If we take, say, $f(n)=\lceil(\log (n+2))\rceil^2$ then the resulting graph therefore has $p_c^\mathrm{site}=0$ but is recurrent by the Nash-Williams criterion since there are $O((\log n)^4)$ edges between the $n$th and $(n+1)$th complete graphs and $\sum_{n=2} (\log n)^{-4}=\infty$. (Note in particular that we have a lot of room in this construction!)
We can make something similar work as a unimodular random rooted graph by using the canopy trees instead of the half-infinite line. This is a pretty standard trick, you can see lots of instances of it in this paper of Omer Angel and myself https://arxiv.org/abs/1710.03003
Indeed, if we take the canopy tree and replace each vertex at height $n$ with a complete graph of size $\lceil (\log (n+2))\rceil^2$ then the resulting graph is recurrent but has $p_c^\mathrm{site}=0$ for similar reasons to the line example. Since the probability that the root of the canopy tree is at height $n$ is $2^{-n}$, we can make this graph unimodular by biasing the height of the root vertex of the canopy tree by $\lceil (\log (n+2))\rceil^2$ then choosing uniformly one of the vertices of the complete graph at that vertex as the root of the new graph (this biasing makes sense since $\lceil (\log (X+2))\rceil^2$ is integrable when $X$ is a geometric random variable). Note that this graph also has rather light tailed degrees: The probability that the root has degree at least $n$ is roughly of order $2^{-\Theta(e^{\sqrt{n})}$.
Basically these examples work because replacing edges by parallel edges (or blowing up vertices into complete graphs) has a much stronger effect on percolation than it does on the random walk.
Best Answer
I don't have anything rigorous to say, but let me share some images that may be useful or interesting to you. The Mathematica code to generate them is here. It's sparsely commented, so feel free to ask in the comments for clarification.
Below, the "out-component" is the set of vertices which are reachable by a directed path from the origin.
Here are a few example out-components at various $p$ in your model for a 161 by 161 grid:
I'm quite fond of this animated GIF file which shows the "averaged" out-component of the vertex at the origin in a 41 by 41 square grid as $p$ is tuned from 0 to 1 (in steps of 0.02). The intensity of a pixel corresponds to the frequency that that vertex was reachable from the origin in a set of 1000 pseudorandom configurations.
I'm not sure what to make of this pattern -- in particular, are they an artifact of the square boundary conditions, as they might cut off longer paths that would have made the dark regions parallel to the $x$ and $y$ axes reachable?
From the same data, here's the probability of percolation (existence of a directed path from the origin to the boundary of that 41 by 41 square grid) as a function of $p$:
And here's the mean fraction of the full grid that is reachable from the origin as a function of $p$:
Perhaps someone with more computer time can run do this with larger system sizes (my run took somewhere around an hour). I might do this for the last two graphs I showed, just to see how the transition sharpens for larger system sizes.
Edit. The last plot doesn't quite tell the full story about the distribution of out-component sizes.
Here's a plot showing the standard deviation of the out-component sizes:
Here's a sequence of plots showing histograms (from 1000 pseudorandomly generated configurations) of the fraction of total vertices reachable from the origin in a 41 by 41 grid at various $p$:
The distribution is bimodal sufficiently near $p=0.5$!
Here's a density plot of the fraction of vertices in the out-component as a function of $p$ -- lighter colors means higher probability density: