The time of bootstrap percolation in two dimensions

In r-neighbour bootstrap percolation on the vertex set of a graph G, vertices are initially infected independently with some probability p. At each time step, the infected set expands by infecting all uninfected vertices that have at least r infected neighbours. We study the distribution of the time t at which all vertices become infected. Given t = t(n) = o(log n/log log n), we prove a sharp threshold result for the probability that percolation occurs by time t in d-neighbou...

In modified two-neighbour bootstrap percolation in two dimensions each site of $\mathbb Z^2$ is initially independently infected with probability $p$ and on each discrete time step one additionally infects sites with at least two non-opposite infected neighbours. In this note we establish that for this model the second term in the asymptotics of the infection time $\tau$ unexpectedly scales differently from the classical two-neighbour model, in which arbitrary two infected ne...

We consider a classic model known as bootstrap percolation on the $n \times n$ square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least $2$ already infected neighbours. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap p...

Answering questions of Itai Benjamini, we show that the event of complete occupation in 2-neighbour bootstrap percolation on the d-dimensional box [n]^d, for d\geq 2, at its critical initial density p_c(n), is noise sensitive, while in k-neighbour bootstrap percolation on the d-regular random graph G_{n,d}, for 2\leq k\leq d-2, it is insensitive. Many open problems remain.

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...

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 ...

In this paper we fill in a fundamental gap in the extremal bootstrap percolation literature, by providing the first proof of the fact that for all $d \geq 1$, the size of the smallest percolating sets in $d$-neighbour bootstrap percolation on $[n]^d$, the $d$-dimensional grid of size $n$, is $n^{d-1}$. Additionally, we prove that such sets percolate in time at most $c_d n^2$, for some constant $c_d >0 $ depending on $d$ only.

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 de...

In this paper, we study the k-neighbor bootstrap percolation process on the d-dimensional grid [n]^d, and show that the minimum number of initial vertices that percolate is (1-d/k)n^d + O(n^{d-1})$ when d<=k<=2d. This confirms a conjecture of Pete.

In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get infected in subsequent rounds if they have at least $r$ infected neighbours. A graph $G$ \emph{percolates} if every vertex becomes eventually infected. A central problem in this process is to determine the critical probability $p_c(G,r)$, at whi...