I hope there exists some answer to this but I am not very good in probability so i do not know how to tackle this problem. It's contemporary-art related so it might not have a good answer. Furthermore, it is clear that this question
What can be said about the number of connected components of $G(n,p)$ random graphs?
is related, however my situation will be quite different.
Begin with the square $X:=[0,1]^2$ and take $n\geq 1$. It is clear that I can divide $X$ into $n^2$ squares, the $(i,j)$-th square being given by :
$$[\frac{i-1}{n},\frac{i}{n}]\times [\frac{j-1}{n},\frac{j}{n}]$$
Let us say we denote $S_{i,j}$ this square. Now take $0\leq p\leq 1$ and take $X_{i,j}$ a random variable which go to $0$ with probability $p$ and to $1$ with probability $1-p$. Of course all the $X_{i,j}$ are taken mutually independent. Then you can construct :
$$Y:=\bigcup_{(i,j)\mid X_{i,j}=1}S_{i,j} $$
Now my question is the following what can be said about the number of connected components of $Y$ ? Even with $p=\frac{1}{2}$, I have trouble finding the law of $|\pi_0(Y)|$...
I have though about the following simplifications one might want to make :
1 The square $S_{i,j}$ is changed to $T_{i,j}$, the square $S_{i,j}$ with its four vertices removed.
Using this a square can be adjacent to at most four squares.
2 Consider the toric $\mathbb{T}$ version of $X$, because in $X$ we get side effects (the square $S_{1,1}$ is in a corner and hence has a very particular role whereas in $\mathbb{T}$ everyone is at the center).
My attempt starts with those two simplifications hence we start from the graph $G_n$ with $n^2$ vertices (each vertex is a square) and each vertices is linked to four vertices (we draw the incidence graph of the CW-complex induced by the squares).
Now the random construction appears, for each vertex there is a probability of $p$ that we delete it with all its contiguous edges. And then I am stuck. I think what I am trying to understand is percolation but it may not be true (do not hesitate to delete the tag if it is not)... Any help will be appreciated.
This is a highly nontrivial question. You should look at the book "Percolation" from Grimmett for more Details, but here are some Facts that are known (which I took from the book). If $K_n$ is the number of conn. comp. (called Clusters) then $K_n/n^2 \to \kappa(p)$ where $\kappa(p)$ is a constant depending on $p$. There is also a central Limit Theorem for $K_n$.