arXiv subjects

Marcos Sá

Publications and source records attributed to Marcos Sá.

5 recordsLinked to original sources

arXiv2026

Near-critical percolation with sparse reinforcements

We study the effect of sparse random reinforcement on near-critical Bernoulli site percolation on the first quadrant of the square lattice. The reinforcement is generated by randomly selecting horizontal and vertical lines and increasing the opening probability $p$ only at their intersections, so that the reinforcement is supported on a sparse set of points with long-range dependence. We determine how the geometry of this random set influences the phase transition. We first investigate whether the reinforcement lowers the critical threshold. When the spacings between consecutive selected lines are geometric, we prove that, for every $p$ above the critical threshold, percolation occurs even if the opening probability $q$ outside the reinforcement is strictly below criticality. In contrast, if the spacings in one direction decay slower than exponentially while those in the other are geometric, then no threshold shift occurs: percolation is impossible for $q$ below the critical threshold, regardless of $p$. We next study the critical case $q=p_c$. We show that sufficiently high moments of the spacing distributions imply percolation at criticality, whereas the absence of a fractional moment of order $α\in(0,1)$ rules it out. Both regimes remain unchanged after an additional independent thinning of the reinforced vertices. Thus, reinforcement supported on random intersections may either lower the critical threshold or leave it unchanged while fundamentally altering the nature of the phase transition.

math.PR↗

arXiv2025

Phase transition on randomly horizontally stretched square lattice

In this article, we study a bond percolation model on a horizontally stretched square lattice, constructed by stretching the distances between the columns of $\mathbb{Z}_+^2$ according to a collection of independent and identically distributed (i.i.d.) copies of a non-negative random variable $ξ$. We assume that $ξ$ satisfies the integrability condition \[ \mathbb{E}\big[ξ\, e^{c(\log ξ)^{1/2}} \,\mathbb{1}_{\{ξ\geq 1\}}\big] < \infty, \] for some constant $c > 8\sqrt{\log 96}$. In this random environment, each vertical edge is independently declared open with probability $p$, while each horizontal edge is open with probability $p^{|e|}$, where $|e|$ denotes the Euclidean length of the edge. We develop a multiscale renormalization scheme adapted to this geometry and use it to prove that percolation occurs for all sufficiently large values of $p < 1$.

math.PR↗

arXiv2023

A new proof for percolation phase transition on stretched lattices

We revisit the phase transition for percolation on randomly stretched lattices. Starting with the usual square grid, keep all vertices untouched while erasing edges according as follows: for every integer $i$, the entire column of vertical edges contained in the line $\{ x = i \}$ is removed independently of other columns with probability $ρ> 0$. Similarly, for every integer $j$, the entire row of horizontal edges contained in the line $\{ y = j\}$ is removed independently with probability $ρ$. On the remaining random lattice, we perform Bernoulli bond percolation. Our main contribution is an alternative proof that the model undergoes a nontrivial phase transition, a result established earlier by Hoffman. The main novelty lies on the fact that the dynamic renormalization employed earlier is replaced by a static version, which is simpler and more robust to extend to different models. We emphasize the flexibility of our methods by showing the non-triviality of the phase transition for a new oriented percolation model in a random environment as well as for a model previously investigated by Kesten, Sidoravicius and Vares. We also prove a result about the sensitivity of the phase transition with respect to the stretching mechanism.

math.PR↗

arXiv2020

Phase transition for percolation on a randomly stretched lattice

Let $\{ξ_i\}_{i \geq 1}$ be a sequence of i.i.d.\ positive random variables. Starting from the usual square lattice replace each horizontal edge that links a site in $i$-th vertical column to another in the $(i+1)$-th vertical column by an edge having length $ξ_i$. Then declare independently each edge $e$ in the resulting lattice open with probability $p_e=p^{|e|}$ where $p\in[0,1]$ and $|e|$ is the length of $e$. We relate the occurrence of nontrivial phase transition for this model to moment properties of $ξ_1$. More precisely, we prove that the model undergoes a nontrivial phase transition when $\mathbb{E}(ξ_1^η)<\infty$, for some $η>1$ whereas, when $\mathbb{E}(ξ_1^η)=\infty$ for some $η<1$, no phase transition occurs.

math.PR↗

arXiv2020

Strict inequality for bond percolation on a dilute lattice with columnar disorder

We consider a dilute lattice obtained from the usual $\mathbb{Z}^3$ lattice by removing independently each of its columns with probability $1-ρ$. In the remaining dilute lattice independent Bernoulli bond percolation with parameter $p$ is performed. Let $ρ\mapsto p_c(ρ)$ be the critical curve which divides the subcritical and supercritical phases. We study the behavior of this curve near the disconnection threshold $ρ_c = p_c^{\text{site}}(\mathbb{Z}^2)$ and prove that, uniformly over $ρ$ it remains strictly below $1/2$ (the critical point for bond percolation on the square lattice $\mathbb{Z}^2)$.

math.PR↗