Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ and a value $0$ with probability $1-p$.

Define a *cluster* of cells as a maximal connected component in the graph of cells with the value of $1$, where edges connect cells whose rows and columns both differ by at most $1$ (so up to eight neighbours for each cell).

I numerically plotted the number of clusters vs. the probability $p$, for matrices of size $8\times 8, 16 \times 16, 32 \times 32, 64 \times 64, 128 \times 128, 256 \times 256.$ The results came out like:

Plot: Number of clusters per unit area vs. probability p for different L's:

(Note that "black cluster" refers to the cluster of elements labelled with $1$)

Is it possible to reformulate this matrix model to match the Erdős–Rényi graph model? I was wondering whether the Giant component result will be applicable for this model and whether it will be possible to reframe this problem to match the Erdős–Rényi model. I think they're related but not sure exactly how. One problem is that the $p$ here is the probability of a node being 1, rather than the probability of the existence of an edge, unlike the Erdős–Rényi model.

This question is a result of Mike's comment, here:

So I am really not sure of the exact probability that there would be a giant component and so I cannot guarantee a giant component likely for the specific value of $p=1/2$ (sorry). But my thinking was along the lines of the analysis of the size of the components of Erdos-Renyi graphs....checking that out may be useful.