Monotone cellular automata in a random environment

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 $\mathbb{Z}^d$.Comment: 33 pages, 7 figure

Cellular Automata and Bootstrap Percolation

We study qualitative properties of two-dimensional freezing cellular automata with a binary state set initialized on a random configuration. If the automaton is also monotone, the setting is equivalent to bootstrap percolation. We explore the extent to which monotonicity constrains the possible asymptotic dynamics by proving two results that do not hold in the subclass of monotone automata. First, it is undecidable whether the automaton almost surely fills the space when initialized on a Bernoulli random configuration with density $p$, for some/all $0 < p < 1$. Second, there exists an automaton whose space-filling property depends on $p$ in a non-monotone way.Comment: 18 pages, 3 figure

Nucleation and growth in two dimensions

We consider a dynamical process on a graph $G$, in which vertices are infected (randomly) at a rate which depends on the number of their neighbours that are already infected. This model includes bootstrap percolation and first-passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph $\mathbb{Z}^2$, significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.Comment: 35 pages, Section 6 update

Complexity, parallel computation and statistical physics

The intuition that a long history is required for the emergence of complexity in natural systems is formalized using the notion of depth. The depth of a system is defined in terms of the number of parallel computational steps needed to simulate it. Depth provides an objective, irreducible measure of history applicable to systems of the kind studied in statistical physics. It is argued that physical complexity cannot occur in the absence of substantial depth and that depth is a useful proxy for physical complexity. The ideas are illustrated for a variety of systems in statistical physics.Comment: 21 pages, 7 figure

Higher order corrections for anisotropic bootstrap percolation

We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics: $$p_c\big( [L]^2,\mathcal{N}_{(1,2)},3 \big) \; = \; \frac{(\log \log L)^2}{12\log L} \, - \, \frac{\log \log L \, \log \log \log L}{ 3\log L} + \frac{\left(\log \frac{9}{2} + 1 \pm o(1) \right)\log \log L}{6\log L}.$$ We note that the second and third order terms are so large that the first order asymptotics fail to approximate $p_c$ even for lattices of size well beyond $10^{10^{1000}}$.Comment: 46 page

Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box

In $r$-neighbour bootstrap percolation, vertices (sites) of a graph $G$ are infected, round-by-round, if they have $r$ neighbours already infected. Once infected, they remain infected. An initial set of infected sites is said to percolate if every site is eventually infected. We determine the maximal percolation time for $r$-neighbour bootstrap percolation on the hypercube for all $r \geq 3$ as the dimension $d$ goes to infinity up to a logarithmic factor. Surprisingly, it turns out to be $\frac{2^d}{d}$, which is in great contrast with the value for $r=2$, which is quadratic in $d$, as established by Przykucki. Furthermore, we discover a link between this problem and a generalisation of the well-known Snake-in-the-Box problem.Comment: 14 pages, 1 figure, submitte