Extremal Bounds for Three-Neighbour Bootstrap Percolation in Dimensions Two and Three

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...

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^3$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,3\}$, where $a_1\le a_2\le a_3$. Suppose we infect any healthy vertex $x\in [L]^3$ already having $a_3+1$ infected neighbours, and that infected sites remain infected forever. In this paper we determine the critical length for percolation up to a...

In 2-neighborhood bootstrap percolation on a graph $G$, an infection spreads according to the following deterministic rule: infected vertices of $G$ remain infected forever and in consecutive rounds healthy vertices with at least two already infected neighbors become infected. Percolation occurs if eventually every vertex is infected. The maximum time $t(G)$ is the maximum number of rounds needed to eventually infect the entire vertex set. In 2013, it was proved by Benevides ...

In this paper we focus on $r$-neighbor bootstrap percolation, which is a process on a graph where initially a set $A_0$ of vertices gets infected. Now subsequently, an uninfected vertex becomes infected if it is adjacent to at least $r$ infected vertices. Call $A_f$ the set of vertices that is infected after the process stops. More formally set $A_t:=A_{t-1}\cup \{v\in V: |N(v)\cap A_{t-1}|\geq r\}$, where $N(v)$ is the neighborhood of $v$. Then $A_f=\bigcup_{t>0} A_t$. We de...

Bootstrap Percolation is a process defined on a graph which begins with an initial set of infected vertices. In each subsequent round, an uninfected vertex becomes infected if it is adjacent to at least $r$ previously infected vertices. If an initially infected set of vertices, $A_0$, begins a process in which every vertex of the graph eventually becomes infected, then we say that $A_0$ percolates. In this paper we investigate bootstrap percolation as it relates to graph dist...

Bootstrap percolation is a type of cellular automaton on graphs, introduced as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be in one of two states: `healthy' or `infected' and from an initial configuration of states, healthy vertices become infected by local rules. While the usual bootstrap processes are monotone in the sets of infected vertices, in this paper, a modification is examined in which infected vertices can return to a healthy state. V...

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^d$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,\dots, d\}$, where $a_1\le a_2\le \dots \le a_d$. Suppose we infect any healthy vertex $v\in [L]^d$ already having $r$ infected neighbours, and that infected sites remain infected forever. In this paper we determine the $(d-1)$-times iterated log...

Majority bootstrap percolation is a monotone cellular automata that can be thought of as a model of infection spreading in networks. Starting with an initially infected set, new vertices become infected once more than half of their neighbours are infected. The average case behaviour of this process was studied on the $n$-dimensional hypercube by Balogh, Bollob\'{a}s and Morris, who showed that there is a phase transition as the typical density of the initially infected set in...

We study a new geometric bootstrap percolation model, line percolation, on the $d$-dimensional integer grid $[n]^d$. In line percolation with infection parameter $r$, infection spreads from a subset $A\subset [n]^d$ of initially infected lattice points as follows: if there exists an axis-parallel line $L$ with $r$ or more infected lattice points on it, then every lattice point of $[n]^d$ on $L$ gets infected, and we repeat this until the infection can no longer spread. The el...

In 2-neighborhood bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least 2 already infected neighbors become infected. Percolation occurs if eventually every vertex is infected. The maximum time t(G) is the maximum number of rounds needed to eventually infect the entire vertex set. In 2013, it was proved \cite{eurocomb13} tha...