This is a sequel to the question: Proof and interpretation of the following percolation theory result for $n\times n$ square grid

In the paper: The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

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?