arXiv subjects

Charles S. do Amaral

Publications and source records attributed to Charles S. do Amaral.

6 recordsLinked to original sources

arXiv2025

Percolation of random compact diamond-shaped systems on the square lattice

We study site percolation on a square lattice with random compact diamond-shaped neighborhoods. Each site $s$ is connected to others within a neighborhood in the shape of a diamond of radius $r_s$, where $r_s$ is uniformly chosen from the set $\{i, i+1, \ldots, m\}$ with $i \leq m$. The model is analyzed for all values of $i = 0, \ldots, 7$ and $m = i, \ldots, 10$, where $\overline{z}(i,m)$ denotes the average number of neighbors per site and $p_c(i,m)$ is the critical percolation threshold. For each fixed $i$, the product $\overline{z}(i,m)\,p_c(i,m)$ is found to converge to a constant as $m \to \infty$. Such behavior is expected when $i=m$ (single diamond sizes), for which the product $\overline{z}(i)\,p_c(i)$ tends toward $2^dη_c$, where $η_c$ is the continuum percolation threshold for diamond-shaped regions or aligned squares in two dimensions ($d=2$). This case is further examined for $i = 1, \ldots, 10$, and the expected convergence is confirmed. The particular case $i = m$ was first studied numerically by Gouker and Family in 1983. We also study the relation to systems of deposited diamond-shaped objects on a square lattice. For monodisperse diamonds of radius $r$, there is a direct mapping to percolation with a diamond-shaped neighborhood of radius $2r+1$, but when there is a distribution of object sizes, there is no such mapping. We study the case of mixtures of diamonds of radius $r=0$ and $r=1$, and compare it to the continuum percolation of disks of two sizes.

cond-mat.stat-mech↗

arXiv2025

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 $s \in \mathbb{Z}^{2}$ 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.

math.PR↗

arXiv2023

Density and correlation in a random sequential adsorption model

We consider a random sequential adsorption process on the one-dimensional lattice with nearest-neighbor exclusion. In this model, each site $s \in \mathbb{Z}$ starts empty and we will try to occupy it in time $t_s$, where $(t_s)_{s\in\mathbb{Z}}$ is a sequence of independent random variables uniformly distributed on the interval $[0,1]$. The site will be occupied if both of its neighbors are vacant. We provide a method to calculate the density of occupied sites up to the time $t$, as well as the pair correlation function.

math.PR↗

arXiv2023

One-dimensional $AB$ random sequential adsorption with one deposition per site

We investigate a modified version of the $AB$ random sequential adsorption model. Specifically, this model involves the deposition of two distinct types of particles onto a lattice, with the constraint that different types cannot occupy neighboring sites. By restricting the deposition attempts to only one per site, we derive an analytical expression for the average densities of particles of types $A$ and $B$, at all time instances, for all deposition probabilities of each particle type.

math-ph↗

arXiv2020

On the monotonicity of the critical time in the Constrained-degree percolation model

The Constrained-degree percolation model was introduced in [B.N.B. de Lima, R. Sanchis, D.C. dos Santos, V. Sidoravicius, and R. Teodoro, Stoch. Process. Appl. (2020)], where it was proven that this model has a non-trivial phase transition on a square lattice. We study the Constrained-degree percolation model on the $d$-dimensional hypercubic lattice ($\mathbb{Z}^d$) and, via numerical simulations, found evidence that the critical time $t_{c}^{d}(k)$ is monotonous not increasing in the constrained $k$ if $d=3,4$, like it is when $d=2$. We verify that the lowest constrained value $k$ such that the system exhibits a phase transition is $k=3$ and that the correlation critical exponent $ν$ for the Constrained-degree percolation model and ordinary Bernoulli percolation are the same.

math-ph↗

arXiv2020

Multirange Ising model on the square lattice

We study the Ising model on $\mathbb{Z}^{2}$ and show, via numerical simulation, that allowing interactions between spins separated by distances $1$ and $m$ (two ranges), the critical temperature, $ T_c (m) $, converges monotonically to the critical temperature of the Ising model on $\mathbb{Z}^4$ as $ m \to \infty $. Only interactions between spins located in directions parallel to each coordinate axis are considered. We also simulated the model with interactions between spins at distances of $ 1 $, $ m $ and $ u $ (three ranges), with $ u $ a multiple of $ m $; in this case our results indicate that $ T_c(m, u) $ converges to the critical temperature of the model on $ \mathbb{Z}^6$. For percolation, analogous results were proven for the critical probability $p_c$ [B. N. B. de Lima, R. P. Sanchis and R. W. C. Silva, Stochastic Process. Appl. {\bf 121}, 2043 (2011)].

cond-mat.stat-mech↗