arXiv subjects

Lucas R. de Lima

Publications and source records attributed to Lucas R. de Lima.

11 recordsLinked to original sources

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↗

arXiv2026

Catastrophe-dispersion models in random and varying environments across generations

We study a class of branching processes in which the offspring distribution is not specified directly but is induced by a cycle of internal colony growth, catastrophic reduction and structured dispersal. The parameters governing growth, survival and dispersal are allowed to vary deterministically or randomly from one generation to the next, giving rise to branching processes in varying and random environments with implicitly defined offspring laws. We show that survival and extinction are governed entirely by the associated log-mean process, exactly as in the classical theory. The paper treats four qualitatively different dispersal mechanisms and establishes a universal ordering of the induced offspring means. For Poissonian growth with binomial survival, explicit thresholds are obtained that determine extinction or survival uniformly over all four mechanisms. A series of ecologically motivated examples with Yule-Simon growth illustrates the versatility of the framework.

math.PR↗

arXiv2025

Infinite-dimensional genetic and evolution algebras generated by Gibbs measures

Genetic and evolution algebras arise naturally from applied probability and stochastic processes. Gibbs measures describe interacting systems commonly studied in thermodynamics and statistical mechanics with applications in several fields. Here, we consider that the algebras are determined by configurations of finite spins on a countable set with their associated Gibbs distributions. The model preserves properties of the finite-dimensional Gibbs algebras found in the literature and extend their results. We introduce infertility in the genetic dynamics when the configurations differ macroscopically. It induces a decomposition of the algebra into a direct sum of fertile ideals with genetic realization. The proposed infinite-dimensional algebras are commutative, non-associative, with uncountable basis and zero divisors. The properties of Gibbs measures allow us to deal with the difficulties arising from the algebraic structure and obtain the results presented in this article.

math-ph↗

arXiv2025

Speed of Convergence and Moderate Deviations of FPP on Random Geometric Graphs

This study delves into first-passage percolation on random geometric graphs in the supercritical regime, where the graphs exhibit a unique infinite connected component. We investigate properties such as geodesic paths, moderate deviations, and fluctuations, aiming to establish a quantitative shape theorem. Furthermore, we examine fluctuations in geodesic paths and characterize the properties of spanning trees and their semi-infinite paths.

math.PR↗

arXiv2025

Coexistence of Species in a Competition Model on Random Geometric Graphs

This paper investigates the coexistence of two competing species on random geometric graphs (RGGs) in continuous time. The species grow by occupying vacant sites according to Richardson's model, while simultaneously competing for occupied sites under the dynamics of the voter model. Coexistence is defined as the event in which both species occupy at least one site simultaneously at any given time. We prove that coexistence occurs with strictly positive annealed probability by applying results from moderate deviations in first-passage percolation and random walk theory, with a focus on specific regions of the space.

math.PR↗

arXiv2024

Asymptotic Shape of Subadditive Processes on Groups and on Random Geometric Graphs

This doctoral thesis undertakes an in-depth exploration of limiting shape theorems across diverse mathematical structures, with a specific focus on subadditive processes within finitely generated groups exhibiting polynomial growth rates, as well as standard First-Passage Percolation (FPP) models applied to Random Geometric Graphs (RGGs). Employing a diverse range of techniques, including subadditive ergodic theorems and tailored modifications suited for polygonal paths within groups, the thesis examines the asymptotic shape under varying conditions. The investigation extends to subadditive cocycles characterized by at least and at most linear growth. Moreover, the study delves into moderate deviations for FPP models on RGGs, refining previous results with theorems that quantify its speed of convergence to the limiting shape, the fluctuation of the geodesics, and its spanning trees. Additionally, we apply the obtained results in a competition model to verify the positive probability of coexistence of two species competing for territory in a random geometric graph.

math.PR↗

arXiv2024

Generic spectrum of the weighted Laplacian operator on Cayley graphs

In this paper, we investigate the spectrum of a class of weighted Laplacians on Cayley graphs and determine under what conditions the corresponding eigenspaces are generically irreducible. Specifically, we analyze the spectrum on left-invariant Cayley graphs endowed with an invariant metric, and we give some criteria for generically irreducible eigenspaces. Additionally, we introduce an operator that is comparable to the Laplacian and show that the same criterion holds.

math.SP↗

arXiv2024

Asymptotic shape for subadditve processes on groups of polynomial growth

This study delves into the exploration of the limiting shape theorem for subadditive processes on finitely generated groups with polynomial growth, commonly referred to as virtually nilpotent groups. Investigating the algebraic structures underlying these processes, we present a generalized form of the asymptotic shape theorem within this framework. Extending subadditive ergodic theory in this context, we consider processes which exhibit both at most and at least linear random growth. We conclude with applications and illustrative examples.

math.PR↗

arXiv2023

Limiting shape for First-Passage Percolation models on Random Geometric Graphs

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.

math.PR↗

arXiv2021

The asymptotic shape theorem for the frog model on finitely generated abelian groups

We study the frog model on Cayley graphs of groups with polynomial growth rate $D \geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.

math.PR↗

arXiv2021

Limit theorems for a random walk with memory perturbed by a dynamical system

We introduce a new random walk with unbounded memory obtained as a mixture of the Elephant Random Walk and the Dynamic Random Walk which we call the Dynamic Elephant Random Walk (DERW). As a consequence of this mixture the distribution of the increments of the resulting random process is time dependent. We prove a strong law of large numbers for the DERW and, in a particular case, we provide an explicit expression for its speed. Finally, we give sufficient conditions for the central limit theorem and the law of the iterated logarithm to hold.

math.PR↗