Imagine turning the square grid defined by $\mathbb{N}^2$ in the plane into a directed graph. The vertices are $\mathbb{N}^2$ and for each vertex $(x,y)$, there is an edge pointing from it to $(x+1, y)$ and another pointing to $(x,y+1)$.
Turn it into a random graph, by letting the edges appear independently at random with probability $p$. Let $p_n$ be the probability that there is a directed path from (0,0) to the set of points of taxi-cab distance $n$ to the origin. Then we see that $p_n$ is a decreasing sequence, whose limit is the probability that there is an infinite directed path.
Take $n$. I want to know how close $p_n$ is to the limit. Let $r_n=p_{n+1}/p_n$. If I know tha t $r_n, r_{n+1}, \dots$ are "large", then somehow, I know I am "close" to the limit. I am trying to come up with a lower bound for $r_n$. The easiest thing, is to argue that, this equals the probability that there is a path reaching points at taxi-cab distance $n+1$ given that there is a path reaching those at distance $n$. This conditional probability is at least $(1-(1-p)^2)$, which is the probability that at one vertex there are edges pointing out from it.
But this is not a good lower bound, because ideally I want to obtain a lower bound that depends on $n$ converging to $1$. Can anyone point out something useful?