arXiv subjects

Roger W. C. Silva

Publications and source records attributed to Roger W. C. Silva.

At least 19 recordsLinked to original sources

arXiv2026

Critical percolation on slabs with random columnar disorder

We explore a bond percolation model on slabs $\mathbb{S}^+_k=\mathbb{Z}_+\times \mathbb{Z}_+\times\{0,\dots,k\}$ featuring one-dimensional inhomogeneities. In this context, a vertical column on the slab comprises the set of vertical edges projecting to the same vertex on $\mathbb{Z}_+\times\{0,\dots,k\}$. Columns are chosen based on the arrivals of a renewal process, where the tail distributions of inter-arrival times follow a power law with exponent $ϕ>1$. Inhomogeneities are introduced as follows: vertical edges on selected columns are open (closed) with probability $q$ (respectively $1-q$), independently. Conversely, vertical edges within unselected columns and all horizontal edges are open (closed) with probability $p$ (respectively $1-p$). We prove that for all sufficiently large $ϕ$ (depending solely on $k$), the following assertion holds: if $q>p_c(\mathbb{S}^+_k)$, then $p$ can be taken strictly smaller than $p_c(\mathbb{S}^+_k)$ in a manner that percolation still occurs.

math.PR↗

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↗

arXiv2026

Truncated long-range percolation of words on the square lattice

We study mixed long-range percolation on the square lattice. Each vertical edge of unit length is independently open with probability $\varepsilon$, and each horizontal edge of length $i$ is independently open with probability $p_i$. Also, each vertex is assigned independently a random variable taking values 1 and 0 with probability $p$ and $1-p$, respectively. We prove that for a broad class of anisotropic long-range percolation models for which connection probabilities $p_i$ satisfy some regularity conditions, all words (semi-infinite binary sequences) are seen simultaneously from the origin with positive probability, even if all edges with length larger than some constant (depending on $\varepsilon$, $p$, and on the sequence $(p_i)$) are suppressed.

math.PR↗

arXiv2026

Sharpness of the phase transition for constrained-degree percolation

We consider constrained-degree percolation on the hypercubic lattice. Initially, all edges are closed, and each edge independently attempts to open at a uniformly distributed random time; the attempt succeeds if, at that instant, both end-vertices have degrees strictly less than a prescribed parameter. The absence of the FKG inequality and the finite energy property, as well as the infinite range of dependency, make the rigorous analysis of the model particularly challenging. In this work, we show that the one-arm probability exhibits exponential decay in its entire subcritical phase. The proof relies on the Duminil-Copin--Raoufi--Tassion randomized algorithm method and resolves a problem of dos Santos and the second author. At the heart of the argument lies an intricate combinatorial transformation of pivotality in the spirit of Aizenman--Grimmett essential enhancements, but with unbounded range. This technique may be of use in other dynamical settings.

math.PR↗

arXiv2025

Constrained-degree percolation on the hypercubic lattice: uniqueness and some of its consequences

We consider the constrained-degree percolation (CDP) model on the hypercubic lattice. This is a continuous-time percolation model defined by a sequence $(U_e)_{e\in\mathcal{E}^d}$ of i.i.d. uniform random variables and a positive integer $k$, referred to as the constraint. The model evolves as follows: each edge $e$ attempts to open at a random time $U_e$, independently of all other edges. It succeeds if, at time $U_e$, both of its end-vertices have degrees strictly smaller than $k$. It is known \cite{hartarsky2022weakly} that this model undergoes a phase transition when $d\geq3$ for most nontrivial values of $k$. In this work, we prove that, for any fixed constraint, the number of infinite clusters at any time $t\in[0,1)$ is almost surely either 0 or 1. This uniqueness result implies the continuity of the percolation function in the supercritical regime, $t\in(t_c,1)$, where $t_c$ denotes the percolation critical threshold. The proof relies on a key time-regularity property of the model: the law of the process is continuous with respect to time for local events. In fact, we establish differentiability in time, thereby extending the result of \cite{SSS} to the CDP setting.

math.PR↗

arXiv2024

Anisotropic Ising model in d+s dimensions

In this note, we consider the asymmetric nearest neighbor ferromagnetic Ising model on the $(d+s)$-dimensional unit cubic lattice $\Z^{d+s}$, at inverse temperature $β=1$ and with coupling constants $J_s>0$ and $J_d>0$ for edges of $\Z^s$ and $\Z^d$, respectively. We obtain a lower bound for the critical curve in the phase diagram of $(J_s,J_d)$. In particular, as $J_d$ approaches its critical value from below, our result is directly related to the so-called dimensional crossover phenomenon.

math-ph↗

arXiv2024

Exponential decay for Constrained-degree percolation

We consider the Constrained-degree percolation model in random environment (CDPRE) on the square lattice. In this model, each vertex $v$ has an independent random constraint $κ_v$ which takes the value $j\in \{0,1,2,3\}$ with probability $ρ_j$. The dynamics is as follows: at time $t=0$ all edges are closed; each edge $e$ attempts to open at a random time $U_e\sim \mathrm{U}(0,1]$, independently of all other edges. It succeeds if at time $U_e$ both its end-vertices have degrees strictly smaller than their respective constraints. We obtain exponential decay of the radius of the open cluster of the origin at all times when its expected size is finite. Since CDPRE is dominated by Bernoulli percolation, such result is meaningful only if the supremum of all values of $t$ for which the expected size of the open cluster of the origin is finite is larger than 1/2. We prove this last fact by showing a sharp phase transition for an intermediate model.

math.PR↗

arXiv2022

Modified Galton-Watson processes with immigration under an alternative offspring mechanism

We propose a novel class of count time series models alternative to the classic Galton-Watson process with immigration (GWI) and Bernoulli offspring. A new offspring mechanism is developed and its properties are explored. This novel mechanism, called geometric thinning operator, is used to define a class of modified GWI (MGWI) processes, which induces a certain non-linearity to the models. We show that this non-linearity can produce better results in terms of prediction when compared to the linear case commonly considered in the literature. We explore both stationary and non-stationary versions of our MGWI processes. Inference on the model parameters is addressed and the finite-sample behavior of the estimators investigated through Monte Carlo simulations. Two real data sets are analyzed to illustrate the stationary and non-stationary cases and the gain of the non-linearity induced for our method over the existing linear methods. A generalization of the geometric thinning operator and an associated MGWI process are also proposed and motivated for dealing with zero-inflated or zero-deflated count time series data.

stat.ME↗

arXiv2021

Constrained-degree percolation in random environment

We consider the Constrained-degree percolation model in random environment on the square lattice. In this model, each vertex $v$ has an independent random constraint $κ_v$ which takes the value $j\in \{0,1,2,3\}$ with probability $ρ_j$. Each edge $e$ attempts to open at a random uniform time $U_e$ in $[0,1]$, independently of all other edges. It succeeds if at time $U_e$ both its end-vertices have degrees strictly smaller than their respectively attached constraints. We show that this model undergoes a non-trivial phase transition when $ρ_3$ is sufficiently large. The proof consists of a decoupling inequality, the continuity of the probability for local events, and a coarse-graining argument.

math.PR↗

arXiv2021

Fractional Poisson random sum and its associated normal variance mixture

In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains the Poisson and geometric summations as particular cases. We show that the weak limit of the FP summation, when properly normalized, is a mixture between the normal and Mittag-Leffler distributions, which we call by Normal-Mittag-Leffler (NML) law. A parameter estimation procedure for the NML distribution is developed and the associated asymptotic distribution is derived. Simulations are performed to check the performance of the proposed estimators under finite samples. An empirical illustration on the daily log-returns of the Brazilian stock exchange index (IBOVESPA) shows that the NML distribution captures better the tails than some of its competitors. Related problems such as a mixed Poisson representation for the FP law and the weak convergence for the Conway-Maxwell-Poisson random sum are also addressed.

math.ST↗

arXiv2021

Exact and computationally efficient Bayesian inference for generalized Markov modulated Poisson processes

Statistical modeling of point patterns is an important and common problem in several areas. The Poisson process is the most common process used for this purpose, in particular, its generalization that considers the intensity function to be stochastic. This is called a Cox process and different choices to model the dynamics of the intensity gives rise to a wide range of models. We present a new class of unidimensional Cox process models in which the intensity function assumes parametric functional forms that switch among them according to a continuous-time Markov chain. A novel methodology is proposed to perform exact Bayesian inference based on MCMC algorithms. The term exact refers to the fact that no discrete time approximation is used and Monte Carlo error is the only source of inaccuracy. The reliability of the algorithms depends on a variety of specifications which are carefully addressed, resulting in a computationally efficient (in terms of computing time) algorithm and enabling its use with large data sets. Simulated and real examples are presented to illustrate the efficiency and applicability of the proposed methodology. A specific model to fit epidemic curves is proposed and used to analyze data from Dengue Fever in Brazil and COVID-19 in some countries.

stat.ME↗

arXiv2020

A note on the dimensional crossover critical exponent

We consider independent anisotropic bond percolation on $\mathbb{Z}^d\times \mathbb{Z}^s$ where edges parallel to $\mathbb{Z}^d$ are open with probability $p<p_c(\mathbb{Z}^d)$ and edges parallel to $\mathbb{Z}^s$ are open with probability $q$, independently of all others. We prove that percolation occurs for $q\geq 8d^2(p_c(\mathbb{Z}^d)-p)$. This fact implies that the so-called Dimensional Crossover critical exponent, if it exists, is greater than 1. In particular, using known results, we conclude the proof that, for $d\geq 11$, the crossover critical exponent exists and equals 1.

math.PR↗

arXiv2020

Convergence and inference for mixed Poisson random sums

In this paper we obtain the limit distribution for partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixing between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between α-stable distributions and NEF laws is established. We propose estimation of the parameters of the NEF models through the method of moments and also by the maximum likelihood method, which is performed via an Expectation-Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators and an empirical illustration on financial market is presented.

math.PR↗

arXiv2020

Multivariate Log-Skewed Distributions with normal kernel and their Applications

We introduce two classes of multivariate log skewed distributions with normal kernel: the log canonical fundamental skew-normal (log-CFUSN) and the log unified skew-normal (log-SUN). We also discuss some properties of the log-CFUSN family of distributions. These new classes of log-skewed distributions include the log-normal and multivariate log-skew normal families as particular cases. We discuss some issues related to Bayesian inference in the log-CFUSN family of distributions, mainly we focus on how to model the prior uncertainty about the skewing parameter. Based on the stochastic representation of the log-CFUSN family, we propose a data augmentation strategy for sampling from the posterior distributions. This proposed family is used to analyze the US national monthly precipitation data. We conclude that a high dimensional skewing function lead to a better model fit.

stat.ME↗

arXiv2017

On a Local Version of the Bak-Sneppen Model

A major difficulty in studying the Bak-Sneppen model is in effectively comparing it with well-understood models. This stems from the use of two geometries: complete graph geometry to locate the global fitness minimizer, and graph geometry to replace the species in the neighborhood of the minimizer. We present a variant in which only the graph geometry is used. This allows to obtain the stationary distribution through random walk dynamics. We use this to show that for constant-degree graphs, the stationary fitness distribution converges to an IID law as the number of vertices tends to infinity. We also discuss exponential ergodicity through coupling, and avalanches for the model.

math.PR↗

arXiv2017

Dimensional Crossover in Anisotropic Percolation on $Z^{d+s}$

We consider bond percolation on $\Z^d\times \Z^s$ where edges of $\Z^d$ are open with probability $p<p_c(\Z^d)$ and edges of $\Z^s$ are open with probability $q$, independently of all others. We obtain bounds for the critical curve in $(p, q)$, with $p$ close to the critical threshold $p_c(\Z^d)$. The results are related to the so-called dimensional crossover from $\Z^d$ to $\Z^{d+s}$.

math.PR↗

arXiv2011

Monotonicity of Percolation Probability on $\Z^d$ with Long Range Connections

Consider an independent site percolation model on $\Z^d,\ d\geq 2$, with parameter $p \in (0,1)$, where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to some coordinate axis. We show that the percolation probability is a non-decreasing function of $k$. We also generalize this result for models whose long range bonds have several lengths.

math.PR↗

arXiv2011

Critical Point and Percolation Probability in a Long Range Site Percolation Model on $\Z^d$

Consider an independent site percolation model with parameter $p \in (0,1)$ on $\Z^d,\ d\geq 2$ where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to each coordinate axis. We show that the percolation threshold of such model converges to $p_c(\Z^{2d})$ when $k$ goes to infinity, the percolation threshold for ordinary (nearest neighbour) percolation on $\Z^{2d}$. We also generalize this result for models whose long range bonds have several lengths.

math.PR↗