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.

  • $\begingroup$ Are you asking for the definition of the percolation threshold? Or are you asking us to read your professor's mind? $\endgroup$ – BebopButUnsteady Nov 10 '11 at 2:18
  • $\begingroup$ I'm asking about how the correlation function is usually defined for percolation on a lattice. $\endgroup$ – Lagerbaer Nov 10 '11 at 2:20
  • $\begingroup$ 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? $\endgroup$ – BebopButUnsteady Nov 10 '11 at 2:42
  • $\begingroup$ 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 $z \ge 3$. $\endgroup$ – Lagerbaer Nov 10 '11 at 2:46

I have no idea what the conventions your class is using, but your confusion comes from assuming that if you pick one point at random, then at percolation it should definitely be part of a very large cluster. This is not true. The number of points which are part of an infinite cluster at percolation vanishes. This means that you only need to have a nonzero probability for an infinite path at any point to ensure that there is an infinite cluster somewhere.

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.

You can solve the problem with any convention for g(r), it's just a translation.


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