Extremal bounds for bootstrap percolation in the hypercube

The r-neighbour bootstrap percolation process on a graph G starts with an initial set A0 of “infected” vertices and, at each step of the process, a healthy vertex becomes infected if it has at least r infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of G eventually becomes infected, then we say that A0 percolates. We prove a conjecture of Balogh and Bollob ́as which says that, for fixed r and d →∞ , every percolating set in the d -dimensional hypercube has cardinality at least 1+ o (1) / r ( d r − 1 ). We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and a related process, known as weak saturation. In addition, we improve on the best known upper bound for the minimum size of a percolating set in the hypercube. In particular, when r = 3, we prove that the minimum cardinality of a percolating set in the d -dimensional hypercube is ⌈ d (d +3) / 6 ⌉ + 1 for all d ≥ 3

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

Saturation in the Hypercube and Bootstrap Percolation

Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $\Theta(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and Computin

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