[Paper Review] Neighbourhood family percolation

Béla Bollobás, Paul Smith+1 more|arXiv (Cornell University)|Apr 18, 2012

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper introduces and analyzes $•$-bootstrap percolation, a general framework for two-state, deterministic, monotone cellular automata on $•^d$ with arbitrary local update rules and random initial configurations. It classifies models into supercritical, critical, and subcritical classes in two dimensions, proving that the critical probability for percolation on $(•/n•)^2$ is $(\log n)^{-\Theta(1)}$ for critical models and $n^{-\Theta(1)}$ for supercritical ones, establishing the first general theory without symmetry assumptions.

ABSTRACT

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 set of infected sites $A_t$ is the union of $A_{t-1}$ and the set of all $x\in\mathbb{Z}^d$ such that $x+X\in A_{t-1}$ for some $X\in\mathcal{U}$. Our model may alternatively be thought of as bootstrap percolation on $\mathbb{Z}^d$ with arbitrary update rules, and for this reason we call it $\mathcal{U}$-bootstrap percolation. In two dimensions, we give a classification of $\mathcal{U}$-bootstrap percolation models into three classes -- supercritical, critical and subcritical -- and we prove results about the phase transitions of all models belonging to the first two of these classes. More precisely, we show that the critical probability for percolation on $(\mathbb{Z}/n\mathbb{Z})^2$ is $(\log n)^{-\Theta(1)}$ for all models in the critical class, and that it is $n^{-\Theta(1)}$ for all models in the supercritical class. The results in this paper are the first of any kind on bootstrap percolation considered in this level of generality, and in particular they are the first that make no assumptions of symmetry. It is the hope of the authors that this work will initiate a new, unified theory of bootstrap percolation on $\mathbb{Z}^d$.

Motivation & Objective

• To develop a general, symmetry-free framework for studying bootstrap percolation on $•^d$ with arbitrary local update rules.
• To classify all two-dimensional $•$-bootstrap percolation models into three universal classes: supercritical, critical, and subcritical.
• To determine the critical probability for percolation in finite systems $(•/n•)^2$ across all models in the supercritical and critical classes.
• To establish the first comprehensive theory of bootstrap percolation that makes no assumptions of symmetry or regularity in update rules.

Proposed method

• Models are defined by a family $• = \{X_1, \dots, X_m\}$ of finite subsets of $•^d \setminus \{\mathbf{0}\}$, representing neighborhood configurations.
• Initial infection occurs via a product measure with density $p$, assigning each site independently to be infected with probability $p$.
• Dynamics evolve deterministically: at each time step, a site becomes infected if any translate of any $X_i \in \u0095$ is fully infected in the previous step.
• The process is monotone: once infected, a site remains infected, and the set of infected sites grows over time.
• The analysis focuses on the critical probability $p_c(n)$ for percolation on the torus $(•/n•)^2$, defined as the threshold where full infection becomes likely.
• Classification into supercritical, critical, and subcritical classes is based on geometric and combinatorial properties of the update family $•$ in two dimensions.

Experimental results

Research questions

• RQ1 What is the critical probability for percolation in $•$-bootstrap percolation on the finite two-dimensional torus $(•/n•)^2$?
• RQ2 How do the critical probabilities scale with $n$ for different classes of update families $•$?
• RQ3 Can a general classification of $•$-bootstrap percolation models be established without assuming symmetry?
• RQ4 What distinguishes supercritical, critical, and subcritical models in terms of their percolation thresholds?
• RQ5 What is the asymptotic behavior of the critical probability for models in the critical and supercritical classes?

Key findings

• For all models in the critical class of $•$-bootstrap percolation in two dimensions, the critical probability on $(•/n•)^2$ is $(\log n)^{-\Theta(1)}$.
• For all models in the supercritical class, the critical probability on $(•/n•)^2$ is $n^{-\Theta(1)}$.
• The classification into supercritical, critical, and subcritical classes is based solely on the geometric and combinatorial structure of the update family $•$ in two dimensions.
• The results are the first to establish phase transition behavior in bootstrap percolation under full generality, without symmetry assumptions.
• The framework unifies and generalizes previous models, providing a foundation for a new, comprehensive theory of bootstrap percolation on $•^d$.