arXiv subjects

Célio Terra

Publications and source records attributed to Célio Terra.

5 recordsLinked to original sources

arXiv2026

On the uniqueness of quasi-stationary distributions for population models with spatial structure

Subcritical population processes are attracted to extinction and do not have non-trivial stationary distributions, which prompts the study of quasi-stationary distributions (QSDs) instead. In contrast to what generally happens for stationary distributions, QSDs may not be unique, even under irreducibility conditions. The general conditions for uniqueness of QSDs are not always easy to check. For the branching process, besides the quasi-limiting distribution there are many other QSDs. In this paper, we investigate whether adding little extra information to the continuous-time branching process is enough to obtain uniqueness. We consider the branching process with genealogy and branching random walks, and show that they have a unique QSD.

math.PR↗

arXiv2026

The modified boundary contact process: invariant measures and critical extinction

We study the one-dimensional modified boundary contact process, in which infections across the two boundary edges of the infected region occur at rate $λ_e$, while all other infections occur at rate $λ_i$. We prove two results in different parts of the phase diagram. First, in the non-attractive region $λ_e>λ_i\geqλ_c$, where $λ_c$ is the critical parameter of the standard contact process, the process seen from its rightmost infected site converges from every semi-infinite initial configuration to an invariant measure. Second, on the critical curve in the attractive region, the infection dies out almost surely.

math.PR↗

arXiv2026

On the slow phase for fixed-energy Activated Random Walks

We study the Activated Random Walk model on the one-dimensional ring, in the high density regime. We develop a toppling procedure that gradually builds an environment that can be used to show that activity will be sustained for a long time. This yields a self-contained and relatively short proof of existence of a slow phase for arbitrarily large sleep rates.

math.PR↗

arXiv2026

Monotonicity of the critical point in two-dimensional oriented percolation with enhancement

In this note, we investigate Bernoulli oriented bond percolation with parameter $p$ on $\mathbb{Z}^2$. In addition to the standard edges, which are open with probability $p$, we introduce diagonal edges each open with probability $\varepsilon$. Every edge is open or closed independently of all other edges. We prove that the critical parameter for this model is strictly decreasing in $\varepsilon$.

math.PR↗

arXiv2024

Dynamic Phenomena in Interacting Particle Systems: Phase Transitions and Equilibrium

This thesis investigates critical phenomena and equilibrium states in various stochastic models through three interconnected studies. In the first chapter, we analyze the Activated Random Walk model on a one-dimensional ring in the high-density regime. We introduce a toppling procedure that incrementally constructs an environment demonstrating the sustained activity over extended periods. This approach provides a concise and self-contained proof of the existence of a slow phase for arbitrarily large sleep rates. The second chapter focuses on a modified unidimensional contact process with varying infection rates. Specifically, infection spreads at rate $λ_e$ at the boundaries of the infected region and at rate $λ_i$ elsewhere. We establish the existence of an invariant measure for this process when $λ_i=λ_c$, $λ_e=λ_c+\varepsilon$ where $λ_c$ denotes the critical parameter for the standard contact process. Furthermore, we demonstrate that the process, when observed from the right edge, converges weakly to this invariant measure. We also show that infection dies almost surely along the critical curve within the attractive region of the phase space. In the final chapter, we explore quasi-stationary distributions (QSDs) for two subcritical population processes in continuous time: branching random walks and branching processes with genealogy. We prove the existence and uniqueness of QSDs for these processes by leveraging spatial aspects of their dynamics.

math.PR↗