Constrained volume-difference site percolation model on the square lattice

We study a percolation model with restrictions on the opening of sites on the square lattice. In this model, each site $s\in Z^2$ starts closed and an attempt to open it occurs at time $t=t_s$, where ${\left(t_s\right)}_{s\in Z^2}$ is a sequence of independent random variables uniformly distributed on the interval $[0,1]$. The site will open if the volume difference between the two largest clusters adjacent to it is greater than or equal to a constant $r$ or if it has at most one adjacent cluster. Through numerical analysis, we determine the critical threshold $t_c\left(r\right)$ for various values of $r$, verifying that $t_c\left(r\right)$ is non-decreasing in $r$ and that there exists a critical value $r_c=5$ beyond which percolation does not occur. Additionally, we find that the correlation length exponent of this model is equal to that of the ordinary percolation model. For $t=1$ and $1\le r\le 9$, we estimate the averages of the density of open sites, the number of distinct cluster volumes, and the volume of the largest cluster.