Bootstrap percolation in high dimensions

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n]^d, for arbitrary functions n = n(t), d = d(t) and r = r(t), as t -> infinity. The main question is to determine the critical probability p_c([n]^d,r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d \gg log n. The bootstrap process has been extensively studied on [n]^d when d is a fixed constant and 2 \leq r \leq d, and in these cases p_c([n]^d,r) has recently been determined up to a factor of 1 + o(1) as n -> infinity. At the other end of the scale, Balogh and Bollobas determined p_c([2]^d,2) up to a constant factor, and Balogh, Bollobas and Morris determined p_c([n]^d,d) asymptotically if d > (log log n)^{2+\eps}, and gave much sharper bounds for the hypercube. Here we prove the following result: let \lambda be the smallest positive root of the equation \sum_{k=0}^\infty (-1)^k \lambda^k / (2^{k^2-k} k!) = 0, so \lambda \approx 1.166. Then (16\lambda / d^2) (1 + (log d / \sqrt{d})) 2^{-2\sqrt{d}} < p_c([2]^d,2) < (16\lambda / d^2) (1 + (5(log d)^2 / \sqrt{d})) 2^{-2\sqrt{d}} if d is sufficiently large, and moreover we determine a sharp threshold for the critical probability p_c([n]^d,2) for every function n = n(d) with d \gg log n.Comment: 51 pages, revised versio

A sharper threshold for bootstrap percolation in two dimensions

Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n x n square, with sites initially infected independently with probability p. The critical probability p_c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim \pi^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining it up to a poly(log log n)-factor. The exponent -3/2 corrects numerical predictions from the physics literature.Comment: 21 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

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

The sharp threshold for bootstrap percolation in all dimensions

In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the d-dimensional grid $[n]^d$. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine $p_c([n]^d,r)$, the density at which percolation (infection of the entire vertex set) becomes likely. In this paper we prove, for every pair $d \ge r \ge 2$, that there is a constant L(d,r) such that $p_c([n]^d,r) = [(L(d,r) + o(1)) / log_(r-1) (n)]^{d-r+1}$ as $n \to \infty$, where $log_r$ denotes an r-times iterated logarithm. We thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. Moreover, we determine L(d,r) for every pair (d,r).Comment: 37 pages, sketch of the proof added, to appear in Trans. of the AM

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