This is a sequel to the question: https://mathoverflow.net/questions/299253/proof-and-interpretation-of-the-following-percolation-theory-result-for-n-times

In the paper: [The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation](https://link.springer.com/article/10.1007%2Fs002200100521)

> Theorem 3.2 basically states that with probability going to $1$ as
> $n\rightarrow\infty$, the largest open cluster in an $n\times n$ box
> is of size $\theta(p)n^2$, $\theta(p)$ being the percolation
> probability. Moreover, Theorem 3.1 part (iii) states that the expectation value of size of the second largest cluster is of sub linear order. 

This result is for bond percolation. However, does there exist any analogous result for site percolation?