Constrained volume-difference site percolation model on the square lattice

Abstract

We study a percolation model with restrictions on the opening of sites on the square lattice. In this model, each site $s \in \mathbb{Z}^{2}$ starts closed and an attempt to open it occurs at time $t=t_s$, where $(t_s)_{s \in \mathbb{Z}^2}$ 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 $t_c(r)$ for various values of $r$, verifying that $t_c(r)$ 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 $1 \leq r \leq 9$, we estimate the averages of the density of open sites, the number of distinct cluster volumes, and the volume of the largest cluster.

Figures (4)

• Figure 1: Part of the square lattice showing the attempt to open site $s$ at time $t=t_s$. All sites marked with the symbols $\bullet$ or $\times$ are open. Sites marked with identical symbols belong to the same cluster. Note that $M_{1}(t_s, s)=|\mathcal{C}_{w}|=12$ and $M_{2}(t_s, s)=|\mathcal{C}_{u}|=|\mathcal{C}_{v}|=9$. Therefore, since $|M_{1}(t_s, s) - M_{2}(t_s, s)| = 3$, the site $s$ will open if and only if the restriction $r$ chosen for the model satisfies $r \leq 3$.
• Figure 2: Graph obtained through a log-log plot using the FSS relation (\ref{['nu_fss']}) for the case $r=1$. The slope of the linear regression corresponds to the estimated value of $\frac{1}{\nu(r)}$.
• Figure 3: Graph obtained through the FSS relation (\ref{['tc']}) to estimate the critical time $t_c(r)$ for the case $r=1$. The linear coefficient of the line obtained from the linear fit is the estimate for $t_c(1)$.
• Figure 4: Graphs of the $\psi_{L}(t;r)$ functions for $r=1$ and all values of $L$. The dashed line indicates the estimated critical time $t_c(r) = 0.633306(4)$. The slope of the graph near the critical time increases as $L$ grows.