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I am trying to learn some stuff about percolation, specifically whatMenshikov'stheorem is about. On wiki http://en.wikipedia.org/wiki/Percolation_theory it says:

"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."

And in http://www.ams.org/samplings/feature-column/fcarc-percolation it says something similar:

"This means that the probability of having an open path decreases exponentially as the distance traveled by the path increases."

Now in the image below:

it seems as though the vast majority of the grid's squares (sites) are members of fairly large clusters. So, if I were to pick a site at random, it seems that chances are it would be in one of these fairly large clusters. However, the wikipedia statement would suggest the randomly picked point would be most likely to be found in a very small cluster (small r), and not one of these large ones.

Can anyone tell me what I am misunderstanding?

Thanks

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# Percolation Theory: Confused

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