# Definition regarding percolation

in a homework sheet studying bond-percolation on the Bethe lattice, a function $g(r)$ is introduced as "the probability of finding two nodes separated by a distance $r$ on the same cluster".

Now just to see if I get this definition right, for the simple one-dimensional case (Bethe lattice with coordination number 2), would this mean that $$g(r) = p^r$$ if $p$ is the probability that a given bond exists, or would it be $$g(r) = 2p^r - p^{2r}$$ The former would mean: Given two nodes of distance $r$, what's the probability that they are connected, whereas the other would mean: Given one node, what's the probability that it's connected to some node a distance $r$ away. And since in the one-dimensional case there are 2 nodes of distance $r$, we get add the probability for having a bond to each of them and then subtract the joint probability...

I'm mainly asking because it will change the answer for the percolation threshold at coordination numbers larger than $2$. If the former definition is correct, I'd guess the correlation function would still be $p^r$ since there's exactly one path between two nodes. Whereas for the latter definition, I'd get something like $$g(r) = 1 - (1-p^r)^{N(r)}$$ where $N(r)$ counts how many nodes exist a distance $r$ away from a chosen "center" node, which is $N(r) = z(z-1)^{n-1}$ if $z$ is the coordination number.

-
I'm asking about how the correlation function is usually defined for percolation on a lattice. – Lagerbaer Nov 10 '11 at 2:20
Sorry didn't mean to sound snarky there. I think the usual definition is the former, the probability that it is connected to a specific node. But the two function should differ only by a polynomial prefactor, no? So how will it change the correlation length or the percolation threshold? – BebopButUnsteady Nov 10 '11 at 2:42
Well, if $g(r) = p^r$, then $g(r) \rightarrow 0$ for $r \rightarrow \infty$ for all $p < 1$, but I know that the percolation threshold is smaller than $1$ for a Bethe lattice with coordination $z \ge 3$. – Lagerbaer Nov 10 '11 at 2:46

The probability g(r) could have been defined in several ways. If you look at a fixed path of length r in the lattice and ask what is the probability that it is connected, you get $p^r$. So a negligible fraction of long paths are connected. But there are (k-1)^r different paths of length r in a k-Bethe Lattice (Cayley graph, homogenous (k-1)-connected graph), so you expect when $p>1/(k-1)$ that there will be a nonzero probability for some infinite path, and if $p<1/2$ then no.