Over the past two installments, we’ve seen that the percolation model, which has at its core simple, static probabilistic rules, exhibits three distinct behaviors. For the occupation probability $p$ well below the critical probability $p_c$, the lattice is mostly empty with a few isolated islands of filled sites but with extremely little likelihood to have a spanning cluster joining opposites of the lattice. For the occupation probability well above the critical probability, the lattice is mostly filled and, as a sort of mirror image, the lattice is mostly filled with a few isolated islands of vacant sites and with nearly perfect certainty that a spanning cluster is present. The third behavior exists in the ‘local’ neighborhood near the critical probability where the observation evidence from computer experiments and the conventional wisdom indicates that many different length sizes exist.
In order to go from that qualitative picture summarized above to a quantitative structure, we need to define the measurements that we can make to the computational lattice. The most important one is the cluster size distribution $n_s$ defined as the number $n$ of clusters with a size $s$.
The best way to understand $n_s$ is with a few sample lattices. For the first example, let’s look at a percolation lattice of size $16 \times 16$ with an occupation probability of $p_{occ} = 0.4$.
**image**
With a bit of patience one might convince oneself that there are 30 distinct clusters. With a bit more effort, it becomes clear that there 13 distinct clusters with only one site and so we would assign $n_s(1) = 13$. Likewise, there are 4 distinct clusters of size 2 so that $n_s(2) = 4$, and so on.
The full listing of Cluster ID versus Cluster Size for this lattice is:
note that a cluster with an ID of zero means that the site was not occupied.