arXiv subjects
Publications and source records attributed to Bernardo N. B. de Lima.
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.
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.
A popular question in Bernoulli percolation models is if the probability of connection between two vertices in a transitive graph decays monotonically with the distance between these two vertices. For example, on the square lattice is an open question to prove that the probability of the origin being connected to the vertex $(0,n)$ is monotone in $n$. In this short note, we exhibit an example of a transitive graph in which the probability of connection between vertices does not necessarily decay as the distance of those vertices grows. We also define a critical point for percolation in $\mathbb{Z}^d$, in which using a generalization of the percolation process it is possible to see the same phenomena happening in the embedding of $\mathbb{Z}^d$ over $\mathbb{R}^d$.
This note was motivated by natural questions related to oriented percolation on a layered environment that introduces long range dependence. As a convenient tool, we are led to deal with questions on the strict decrease of the percolation parameter in the oriented setup when an extra dimension is added.
We consider a dependent percolation model on the square lattice $\mathbb{Z}^2$. The range of dependence is infinite in vertical and horizontal directions. In this context, we prove the existence of a phase transition. The proof exploits a multi-scale renormalization argument that is defined once the environment configuration is suitably good and, which, together with the main estimate for the induction step, comes from Kesten, Sidoravicius and Vares (To appear in {\em Electronic Journal of Probability}, (2022)). This work was inspired by de Lima (Ph.D.Thesis, \emph{Informes de Matemática. IMPA}, Série C-26/2004) where the simpler case of a deterministic environment was considered. It has various applications, including an alternative proof for the phase transition on the two dimensional random stretched lattice proved by Hoffman ({\em Comm. Math. Phys.} {\bf 254}, 1-22 (2005)).
We consider some problems related to the truncation question in long-range percolation. It is given probabilities that certain long-range oriented bonds are open; assuming that this probabilities are not summable, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. This question is still open if the set of vertices is $\Z^2$. We give some conditions in which the answer is affirmative. One of these results generalize the previous result in [Alves, Hilário, de Lima, Valesin, Journ. Stat. Phys. {\bf 122}, 972 (2017)].
We study the phase transition phenomena for long-range oriented percolation and contact process. We studied a contact process in which the range of each vertex are independent, updated dynamically and given by some distribution $N$. We also study an analogous oriented percolation model on the hyper-cubic lattice, here there is a special direction where long-range oriented bonds are allowed; the range of all vertices are given by an i.i.d. sequence of random variables with common distribution $N$. For both models, we prove some results about the existence of a phase transition in terms of the distribution $N$.
Let $ \mathbb{L}^{d} = ( \mathbb{Z}^{d},\mathbb{E}^{d} ) $ be the $ d $-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on $ \mathbb{L}^{d} $ in which every edge inside the $ s $-dimensional hyperplane $ \mathbb{Z}^{s} \times \{ 0 \}^{d-s} $, $ 2 \leq s < d $, is open with probability $ q $ and every other edge is open with probability $ p $. We prove the uniqueness of the infinite cluster in the supercritical regime whenever $ p \neq p_{c}(d) $, where $ p_{c}(d) $ denotes the threshold for homogeneous percolation, and that the critical point $ (p,q_{c}(p)) $ can be approximated on the phase space by the critical points of slabs, for any $ p < p_{c}(d) $.
We consider an inhomogeneous oriented percolation model introduced by de Lima, Rolla and Valesin. In this model, the underlying graph is an oriented rooted tree in which each vertex points to each of its $d$ children with `short' edges, and in addition, each vertex points to each of its $d^k$ descendant at a fixed distance $k$ with `long' edges. A bond percolation process is then considered on this graph, with the prescription that independently, short edges are open with probability $p$ and long edges are open with probability $q$. We study the behavior of the critical curve $q_c(p)$: we find the first two terms in the expansion of $q_c(p)$ as $k \to \infty$, and prove that the critical curve lies strictly above the critical curve of a related branching process, in the relevant parameter region. We also prove limit theorems for the percolation cluster in the supercritical, subcritical and critical regimes.
We present large deviations estimates in the supremum norm for a system of independent random walks superposed with a birth-and-death dynamics evolving on the discrete torus with $N$ sites. The scaling limit considered is the so-called \textit{high density limit} (see the survey \cite{franco} on the subject), where space, time and initial quantity of particles are rescaled. The associated rate functional here obtained is a semi-linearised version of the rate function of \cite{JonaLandimVares}, which dealt with large deviations of exclusion processes superposed with birth-and-death dynamics. An ingredient in the proof of large deviations consists in providing a limit of a suitable class of perturbations of the original process. This is precisely one of the main contributions of this work: a strategy to extend the original high density approach (as in \cite{Arnold,blount2,blount,francogroisman,Kote2,KoteHigh1988}) to weakly asymmetric systems. Two cases are considered with respect to the initial quantity of particles, the power law and the (at least) exponential growth. In the first case, we present the lower bound only on a certain subset of smooth profiles, while in the second case, additionally assuming concavity of the birth and the death functions and a constant initial profile, we provide a full large deviations principle.
We propose a method based on cluster expansion to study the low activity/high temperature phase of a continuous particle system confined in a finite volume, interacting through a stable and finite range pair potential with negative minimum in presence of non free boundary conditions.
We consider the Constrained-degree percolation model on the hypercubic lattice, $\mathbb L^d=(\mathbb Z^d,\mathbb E^d)$ for $d\geq 3$. It is a continuous time percolation model defined by a sequence, $(U_e)_{e\in\mathbb E^d}$, of i.i.d. uniform random variables in $[0,1]$ and a positive integer (constraint) $κ$. Each bond $e\in\mathbb E^d$ tries to open at time $U_e$; it succeeds if and only if both its end-vertices belong to at most $κ-1$ open bonds at that time. Our main results are quantitative upper bounds on the critical time, characterising a phase transition for all $d\geq 3$ and most nontrivial values of $κ$. As a byproduct, we obtain that for large constraints and dimensions the critical time is asymptotically $1/(2d)$. For most cases considered it was previously not even established that the phase transition is nontrivial. One of the ingredients of our proof is an improved upper bound for the critical curve, $s_{\mathrm{c}}(b)$, of the Bernoulli mixed site-bond percolation in two dimensions, which may be of independent interest.
The Constrained-degree percolation model was introduced in [B.N.B. de Lima, R. Sanchis, D.C. dos Santos, V. Sidoravicius, and R. Teodoro, Stoch. Process. Appl. (2020)], where it was proven that this model has a non-trivial phase transition on a square lattice. We study the Constrained-degree percolation model on the $d$-dimensional hypercubic lattice ($\mathbb{Z}^d$) and, via numerical simulations, found evidence that the critical time $t_{c}^{d}(k)$ is monotonous not increasing in the constrained $k$ if $d=3,4$, like it is when $d=2$. We verify that the lowest constrained value $k$ such that the system exhibits a phase transition is $k=3$ and that the correlation critical exponent $ν$ for the Constrained-degree percolation model and ordinary Bernoulli percolation are the same.
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.
In the Constrained-degree percolation model on a graph $(\mathbb{V},\mathbb{E})$ there are a sequence, $(U_e)_{e\in\mathbb{E}}$, of i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$. Each bond $e$ tries to open at time $U_e$, it succeeds if both its end-vertices would have degrees at most $k-1$. We prove a phase transition theorem for this model on the square lattice $\mathbb{L}^2$, as well as on the d-ary regular tree. We also prove that on the square lattice the infinite cluster is unique in the supercritical phase.
We establish a connection between the entropy compression method and the Moser-Tardos algorithmic version of the Lovász local lemma through the cluster expansion of the subset gas. We also show that the Moser-Tardos resampling algorithm and the entropy compression bactracking algorithm produce identical bounds.
In this note we obtain a new bound for the acyclic edge chromatic number $a'(G)$ of a graph $G$ with maximum degree $D$ proving that $a'(G)\leq 3.569(D-1)$. To get this result we revisit and slightly modify the method described in [Giotis, Kirousis, Psaromiligkos and Thilikos, Theoretical Computer Science, 66: 40-50, 2017].
Let $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $\mathbb{Z}$. We choose a collection $\mathcal{C}$ of finite connected subgraphs of $G$ and consider a model of Bernoulli bond percolation on $\mathbb{G}$ which assigns probability $q$ of being open to each edge whose projection onto $G$ lies in some subgraph of $\mathcal{C}$ and probability $p$ to every other edge. We show that the critical percolation threshold $p_{c}\left(q\right)$ is a continuous function in $\left(0,1\right)$, provided that the graphs in $\mathcal{C}$ are "well-spaced" in $G$ and their vertex sets have uniformly bounded cardinality. This generalizes a recent result due to Szabó and Valesin.