# Percolation Theory: Confused

by ddriver1
Tags: confused, percolation, theory
 P: 3 Hi, I am trying to learn some stuff about percolation, specifically what Menshikov's theorem is about. On wiki http://en.wikipedia.org/wiki/Percolation_theory it says: "when p
 Mentor P: 11,576 The large clusters are just more visible (as they are larger :D). In addition, I think this exponential decay is the limit for large cluster sizes. The closer p is to pc, the larger the cluster sizes where you see this exponential decay.
P: 3
 Quote by mfb The large clusters are just more visible (as they are larger :D). In addition, I think this exponential decay is the limit for large cluster sizes. The closer p is to pc, the larger the cluster sizes where you see this exponential decay.
Hmm. But the large clusters are taking up a visibly large amount of area. So if for example the area of the whole lattice is 1000x1000 = 1000000, and one of the clusters is of size 300000, then there is
a 30% chance that an arbitrarily picked site is in a cluster with size 300000. Maybe the wiki definition is saying that there is close to 0% chance of it being in a cluster of size exactly 300000 (as opposed to say, 300001)? If so, it kind of makes sense when you consider that the image shown is just one specific instance.

But then the second definition confuses me, as surely if most of the clusters are fairly large, then the chances of there being a path of distance 1 is going to be almost the same as the chances of a path of distance 2. Would it be possible that by "decreases exponentially" he means only after a certain value of r, like in the graph i've attached?

 HW Helper P: 3,440 Percolation Theory: Confused I think in the image, it is just easier to see the larger clusters as mfb says. It seems possible to me that truly the cluster size does obey exponential decay, it is just hard to see it. In Kesten's paper, the theorem (as he wrote it) is like this: For any ##p<1/2## there exists a constant ##C_1(p)>0## such that for all n $$P_p ( \text{W contains vertices at a distance} \geq \text{n from the origin} ) \leq 2e^{-C_1(p)n}$$ And he explains earlier that ##W## is the open cluster that contains the origin. So, it really does seem like it is an exponential decay, for any distance from the origin. So (I think) even the small clusters obey this exponential decay law. Well, it is an inequality, not truly an equation giving the exact value of the probability. edit: also, since it is saying the probability is less than or equal to that amount, it means that the decay is at least exponential. So we are guaranteed that the probability of size of cluster decays at least exponentially with the cluster size. In other words, it could decay even faster, but not slower, than exponential.
P: 3
 Quote by BruceW I think in the image, it is just easier to see the larger clusters as mfb says. It seems possible to me that truly the cluster size does obey exponential decay, it is just hard to see it. In Kesten's paper, the theorem (as he wrote it) is like this: For any ##p<1/2## there exists a constant ##C_1(p)>0## such that for all n $$P_p ( \text{W contains vertices at a distance} \geq \text{n from the origin} ) \leq 2e^{-C_1(p)n}$$ And he explains earlier that ##W## is the open cluster that contains the origin. So, it really does seem like it is an exponential decay, for any distance from the origin. So (I think) even the small clusters obey this exponential decay law. Well, it is an inequality, not truly an equation giving the exact value of the probability. edit: also, since it is saying the probability is less than or equal to that amount, it means that the decay is at least exponential. So we are guaranteed that the probability of size of cluster decays at least exponentially with the cluster size. In other words, it could decay even faster, but not slower, than exponential.
You are right, if we say have 1000 clusters in total, we will have many clusters of size 1, fewer clusters of size 2, fewer sizes of size 3 etc. But of course there will be a few very large clusters.
So the text I quoted from Wiki:

"when p<pc, the probability that a specific point (for example, the origin) is contained in an open cluster of size r decays to zero exponentially in r."

makes sense, as the probability of a site being in a cluster of size 1 is much higher than it being in
a cluster of size, say, 26845.

Now that you rephrased Kesten's paper, it seems like you are saying that sites near the origin are very likely to be members of the cluster, and as you consider vertices which are further away they are exponentially less likely to be in that cluster. This would make sense, and would apply to any cluster, regardless of its size.

However, in that case the two statements are saying different things, and they are both supposed to be based off Menshikov's theorem (or are they?) One is saying that you get many small clusters and fewer large clusters, while the theorem you mentioned is talking about points in clusters and not anything to do with cluster sizes themselves.