Constrained volume-difference site percolation model on the square lattice
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915141810913280 |
|---|---|
| author | Amaral, Charles S. do |
| author_facet | Amaral, Charles S. do |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04409 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constrained volume-difference site percolation model on the square lattice Amaral, Charles S. do Probability Mathematical Physics 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. |
| title | Constrained volume-difference site percolation model on the square lattice |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2408.04409 |