ID: 2303.13920

Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules

March 24, 2023

View on ArXiv

Similar papers 2

Universality for two-dimensional critical cellular automata

June 25, 2014

85% Match
Béla Bollobás, Hugo Duminil-Copin, ... , Smith Paul
Probability
Combinatorics

We study the class of monotone, two-state, deterministic cellular automata, in which sites are activated (or 'infected') by certain configurations of nearby infected sites. These models have close connections to statistical physics, and several specific examples have been extensively studied in recent years by both mathematicians and physicists. This general setting was first studied only recently, however, by Bollob\'as, Smith and Uzzell, who showed that the family of all su...

Find SimilarView on arXiv

The second term for two-neighbour bootstrap percolation in two dimensions

June 23, 2018

85% Match
Ivailo Hartarsky, Robert Morris
Probability
Combinatorics

In the $r$-neighbour bootstrap process on a graph $G$, vertices are infected (in each time step) if they have at least $r$ already-infected neighbours. Motivated by its close connections to models from statistical physics, such as the Ising model of ferromagnetism, and kinetically constrained spin models of the liquid-glass transition, the most extensively-studied case is the two-neighbour bootstrap process on the two-dimensional grid $[n]^2$. Around 15 years ago, in a major ...

Find SimilarView on arXiv

Bootstrap percolation, probabilistic cellular automata and sharpness

December 3, 2021

85% Match
Ivailo Hartarsky
Probability
Statistical Mechanics
Cellular Automata and Lattic...

We establish new connections between percolation, bootstrap percolation, probabilistic cellular automata and deterministic ones. Surprisingly, by juggling with these in various directions, we effortlessly obtain a number of new results in these fields. In particular, we prove the sharpness of the phase transition of attractive absorbing probabilistic cellular automata, a class of bootstrap percolation models and kinetically constrained models. We further show how to recover a...

Find SimilarView on arXiv

The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions

March 28, 2006

84% Match
Alexander E. Holroyd
Probability
Mathematical Physics

In the modified bootstrap percolation model, sites in the cube {1,...,L}^d are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the d dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all d>=2 we prove that as L\to\infty and p\to 0 simultaneously, this probability converges to ...

Find SimilarView on arXiv

Subcritical bootstrap percolation via Toom contours

March 30, 2022

84% Match
Ivailo Hartarsky, Réka Szabó
Probability
Combinatorics

In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions, but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshol...

Find SimilarView on arXiv

Abelian Cascade Dynamics in Bootstrap Percolation

July 27, 1998

84% Match
S. S. Manna
Disordered Systems and Neura...
Soft Condensed Matter

The culling process in Bootstrap Percolation is Abelian since the final stable configuration does not depend on the details of the updating procedure. An efficient algorithm is devised using this idea for the determination of the bootstrap percolation threshold in two dimension which takes $L^2$ time compared to the $L^3 \log L$ in the conventional method. A generalised Bootstrap Percolation allowing many particles at a site is studied where continuous phase transitions are o...

Find SimilarView on arXiv

Monotone cellular automata in a random environment

April 18, 2012

84% Match
Béla Bollobás, Paul Smith, Andrew Uzzell
Probability
Combinatorics

In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in $\mathbb{Z}^d$ with random initial configurations. Formally, we are given a set $\mathcal{U}=\{X_1,\dots,X_m\}$ of finite subsets of $\mathbb{Z}^d\setminus\{\mathbf{0}\}$, and an initial set $A_0\subset\mathbb{Z}^d$ of `infected' sites, which we take to be random according to the product measure with density $p$. At time $t\in\mathbb{N}$, the...

Find SimilarView on arXiv

Recent advances and open challenges in percolation

April 21, 2014

84% Match
N. A. M. Araújo, P. Grassberger, B. Kahng, ... , Ziff R. M.
Statistical Mechanics

Percolation is the paradigm for random connectivity and has been one of the most applied statistical models. With simple geometrical rules a transition is obtained which is related to magnetic models. This transition is, in all dimensions, one of the most robust continuous transitions known. We present a very brief overview of more than 60 years of work in this area and discuss several open questions for a variety of models, including classical, explosive, invasion, bootstrap...

Find SimilarView on arXiv

Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation

October 21, 2004

84% Match
Federico Camia
Probability
Statistical Mechanics
Mathematical Physics

Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$ where the initial configuration $\omega_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least three neighboring 1's. In this paper we show that the configuration $\omega_n$ at time n converges exponentially fast to a final configuration $\bar\omega$, and that the limiting measure corresponding to $\bar\omega$ is in the universality c...

Find SimilarView on arXiv

The sharp threshold for the Duarte model

March 16, 2016

83% Match
Béla Bollobás, Hugo Duminil-Copin, ... , Smith Paul
Probability
Combinatorics

The class of critical bootstrap percolation models in two dimensions was recently introduced by Bollob\'as, Smith and Uzzell, and the critical threshold for percolation was determined up to a constant factor for all such models by the authors of this paper. Here we develop and refine the techniques introduced in that paper in order to determine a sharp threshold for the Duarte model. This resolves a question of Mountford from 1995, and is the first result of its type for a mo...

Find SimilarView on arXiv