I am trying to learn some stuff about percolation. On wiki (http://en.wikipedia.org/wiki/Percolation_theory) it says:
"when $p<p_{c}$, the probability that a specific point (for example, the origin) is contained in an open cluster of size $r$ decays to zero exponentially in $r$."
... which suggests that we are very likely to be in a small cluster, but less and less likely to be in an increasingly large cluster. (I think)
I am slightly confused by this. In the picture below are clusters on a $1000$x$1000$ square lattice (site percolation, where $p_{c}=0.592...$) and the image was made when $p=0.58$, so $p<p_{c}$.
However, if you look at the image, most of the area is taken up by large clusters. So if we take a specific point on the grid, surely we're most likely to find that it is in one of these large clusters?
Can anyone explain what I am misunderstanding here?
I think the confusion comes from the fact that the event '0 is in a large cluster' is not the same (and has a smaller probability of) as 'the size of a cluster when randomly choosing a point', because you will have more chance of falling into a large cluster, if you choose randomly. It's like when you are waiting for the train that follows a poisson with intensity lambda. when you arrive randomly you might think you will wait for lambda / 2 but actually you will have to wait for lambda because you have more chance to get there when the 2 train are far from each other. (hope i am clear)