Constrained volume-difference site percolation model on the square lattice

TL;DR

This paper introduces a constrained site percolation model on the square lattice, analyzing its critical thresholds and properties, and compares its behavior to ordinary percolation through numerical simulations.

Contribution

The study defines a new constrained percolation model with volume difference restrictions and determines its critical thresholds and exponents via numerical analysis.

Findings

Critical threshold $t_c(r)$ increases with $r$

Percolation ceases beyond $r_c=5$

Correlation length exponent matches ordinary percolation

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.