arXiv subjects

Gustavo O. de Carvalho

Publications and source records attributed to Gustavo O. de Carvalho.

5 recordsLinked to original sources

arXiv2026

Phase transitions for percolation of words in one dimension

In this paper, we investigate models of percolation of words on $\mathbb{Z}_+$ with long-range connections. The underlying graph is oriented and constructed according to a sequence of non-negative ranges. In the first model, the sequence of ranges is considered random and, in the second model, deterministic and non-decreasing. In both cases, we establish a phase transition for the occurrence of percolation of all words simultaneously from the origin.

math.PR↗

arXiv2026

Improved Survival Results for the One-Dimensional Renewal Contact Process

The renewal contact process is a non-Markovian variant of the classical contact process in which recoveries are governed by independent renewal processes with interarrival distribution $μ$. We establish new sufficient conditions ensuring finiteness of the critical infection parameter $λ_c(μ)$ for the one-dimensional model. In particular, we prove that $λ_c(μ)<+\infty$ for every non-degenerate arithmetic interarrival distribution. Moreover, finiteness holds whenever the atomic component of the renewal measure is uniformly small on sufficiently short intervals. This criterion applies in particular to all non-atomic interarrival distributions, including singular continuous laws. The proof combines local estimates for renewal measures with a comparison to a regenerative oriented percolation model and a Peierls-type contour argument.

math.PR↗

arXiv2025

Frog model on $\mathbb{Z}$ with random survival parameter

We study the frog model on $ \mathbb{Z} $ with geometric lifetimes, introducing a random survival parameter. Active and inactive particles are placed at the vertices of $ \mathbb{Z} $. The lifetime of each active particle follows a geometric random variable with parameter $ 1-p $, where $ p $ is randomly sampled from a distribution $ π$. Each active particle performs a simple random walk on $ \mathbb{Z} $ until it dies, activating any inactive particles it encounters along its path. In contrast to the usual case where $ p $ is fixed, we show that there exist non-trivial distributions $ π$ for which the model survives with positive probability. More specifically, for $π\sim Beta(α,β)$, we establish the existence of a critical value $ β=0.5 $, that separates almost sure extinction from survival with positive probability. Furthermore, we show that the model is recurrent whenever it survives with positive probability.

math.PR↗

arXiv2024

Critical Conditions for the Coverage of Complete Graphs with the Frog Model

We consider a system of interacting random walks known as the frog model. Let $\mathcal{K}_n=(\mathcal{V}_n,\mathcal{E}_n)$ be the complete graph with $n$ vertices and $o\in\mathcal{V}_n$ be a special vertex called the root. Initially, $1+η_o$ active particles are placed at the root and $η_v$ inactive particles are placed at each other vertex $v\in\mathcal{V}_n\setminus\{o\}$, where $\{η_v\}_{v\in \mathcal{V}_n}$ are i.i.d. random variables. At each instant of time, each active particle may die with probability $1-p$. Every active particle performs a simple random walk on $\mathcal{K}_n$ until the moment it dies, activating all inactive particles it hits along its path. Let $V_\infty(\mathcal{K}_n,p)$ be the total number of visited vertices by some active particle up to the end of the process, after all active particles have died. In this paper, we show that $V_\infty(\mathcal{K}_n,p_n)\geq (1-ε)n$ with high probability for any fixed $ε>0$ whenever $p_n\rightarrow 1$. Furthermore, we establish the critical growth rate of $p_n$ so that all vertices are visited. Specifically, we show that if $p_n=1-\fracα{\log n}$, then $V_\infty(\mathcal{K}_n,p_n)=n$ with high probability whenever $0<α E(η)$.

math.PR↗

arXiv2022

The coverage ratio of the frog model on complete graphs

The frog model is a system of interacting random walks. Initially, there is one particle at each vertex of a connected graph $\mathcal{G}$. All particles are inactive at time zero, except for the one which is placed at the root of $\mathcal{G}$, which is active. At each instant of time, each active particle may die with probability $1-p$. Once an active particle survives, it jumps on one of its nearest vertices, chosen with uniform probability, performing a discrete time simple symmetric random walk (SRW) on $\mathcal{G}$. Up to the time it dies, it activates all inactive particles it hits along its way. From the moment they are activated on, every such particle starts to walk, performing exactly the same dynamics, independent of everything else. In this paper, we take $\mathcal{G}$ as the $n-$complete graph ($\mathcal{K}_n$, a finite graph with each pair of vertices linked by an edge). We study the limit in $n$ of the coverage ratio, that is, the proportion of visited vertices by some active particle up to the end of the process, after all active particles have died.

math.PR↗