This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix?
In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of size $1$ clusters in a random matrix are always greater than the number of sizes $2$ clusters in large $N\times N$ matrices (for any $p\in (0,1)$). From my data files, it also seems that the number of sizes $2$ clusters will be greater than the number of size $3$ clusters for all $p\in (0,1)$. That is, at least for the first few natural numbers $n$, the number of clusters of size $n$ is greater than the number of clusters of size $n+1$. Around the site percolation threshold $p=0.407$ there seem to be some fluctuations, however, still, for the first few natural numbers, the cluster sizes continue showing the above trend.
So, my question basically is: Is it possible to generalize the above trend? If yes, up to which natural number $n$ can it be generalized, and why?
@SylvainJULIEN made an interesting comment:
This sounds a bit like some kind of graphic Benford's law.
I'm not sure if Benford's law is somehow applicable in this situation. However, I'd be interested to hear if someone has any idea regarding this.