arXiv subjects

Marcelo R. Hilário

Publications and source records attributed to Marcelo R. Hilário.

8 recordsLinked to original sources

arXiv2026

The extinction of the contact process in a one-dimensional random environment with long-range interactions

We study the contact process on the long-range percolation cluster on $\mathbb{Z}$ where each edge $\langle i,j \rangle$ is open with probability $|i-j|^{-s}$ for $s> 2$. Using a renormalization procedure we apply Peierls-type argument to prove that the contact process dies out if the transmission rate is smaller than a critical threshold. Our methods involve the control of crossing probabilities for percolation on randomly-stretched lattices as in https://doi.org/10.1214/22-AAP1887.

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↗

arXiv2023

Law of large numbers for ballistic random walks in dynamic random environments under lateral decoupling

We establish a strong law of large numbers for one-dimensional continuous-time random walks in dynamic random environments under two main assumptions: the environment is required to satisfy a decoupling inequality that can be interpreted as a bound on the speed of dependence propagation, while the random walk is assumed to move ballistically with a speed larger than this bound. Applications include environments with strong space-time correlations such as the zero-range process and the asymmetric exclusion process.

math.PR↗

arXiv2020

Bernoulli Hyperplane Percolation

We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.

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↗

arXiv2019

Random walk on the simple symmetric exclusion process

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density $ρ\in [0, 1]$ of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $ρ$ except for at most two values $ρ_-, ρ_+ \in [0, 1]$. The asymptotic speed we obtain in our LLN is a monotone function of $ρ$. Also, $ρ_-$ and $ρ_+$ are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is $1/2$ and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. Finally, we prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

math.PR↗

arXiv2015

Bernoulli line percolation

We introduce a percolation model on $\mathbb{Z}^d$, $d \geq 3$, in which the discrete lines of vertices that are parallel to the coordinate axis are entirely removed at random and independently of each other. In this way a vertex belongs to the vacant set $\mathcal{V}$ if and only if none of the $d$ lines to which it belongs, is removed. We show the existence of a phase transition for $\mathcal{V}$ as the probability of removing the lines is varied. We also establish that, in the certain region of parameters space where $\mathcal{V}$ contains an infinite component, the truncated connectivity function has power-law decay, while inside the region where $\mathcal{V}$ has no infinite component, there is a transition from exponential to power-law decay. In the particular case $d=3$ the power-law decay extends through all the region where $\mathcal{V}$ has an infinite connected component. We also show that the number of infinite connected components of $\mathcal{V}$ is either $0$, $1$ or $\infty$.

math.PR↗

arXiv2013

Embedding binary sequences into Bernoulli site percolation on $\mathbb{Z}^3$

We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on $\mathbb{Z}^d$ with parameter $p$, known also as percolation of words.\ In 1995, I.\ Benjamini and H.\ Kesten proved that, for $d \geq 10$ and $p=1/2$, all sequences can be embedded, almost surely. They conjectured that the same should hold for $d \geq 3$. In this paper we consider $d \geq 3$ and $p \in (p_c(d), 1-p_c(d))$, where $p_c(d)<1/2$ is the critical threshold for site percolation on $\mathbb{Z}^d$. We show that there exists an integer $M = M (p)$, such that, a.s., every binary sequence, for which every run of consecutive {0s} or {1s} contains at least $M$ digits, can be embedded.

math.PR↗