The time of bootstrap percolation with dense initial sets for all thresholds

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 tu...

The Hamming torus of dimension $d$ is the graph with vertices $\{1,\dots,n\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $\theta$ starts with a random set of open vertices, to which every vertex belongs independently with probability $p$, and at each time step the open set grows by adjoining every vertex with at least $\theta$ open neighbors. We assume that $n$ is large and that $p$ scales as $n^{-\alpha}$ f...

In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain infected forever. Percolation occurs if eventually every vertex is infected. The elements of the set of initially infected vertices, A \subset V(G), are normally chosen independently at random, each with probability p, say. This process ...

Graph bootstrap percolation, introduced by Bollob\'as in 1968, is a cellular automaton defined as follows. Given a "small" graph $H$ and a "large" graph $G = G_0 \subseteq K_n$, in consecutive steps we obtain $G_{t+1}$ from $G_t$ by adding to it all new edges $e$ such that $G_t \cup e$ contains a new copy of $H$. We say that $G$ percolates if for some $t \geq 0$, we have $G_t = K_n$. For $H = K_r$, the question about the size of the smallest percolating graphs was independe...

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...

We investigate the behaviour of $r$-neighbourhood bootstrap percolation on the binomial $k$-uniform random hypergraph $H_k(n,p)$ for given integers $k\geq 2$ and $r\geq 2$. In $r$-neighbourhood bootstrap percolation, infection spreads through the hypergraph, starting from a set of initially infected vertices, and in each subsequent step of the process every vertex with at least $r$ infected neighbours becomes infected. For our analysis the set of initially infected vertices i...

This paper analyzes various questions pertaining to bootstrap percolation on the $d$-dimensional Hamming torus where each node is open with probability $p$ and the percolation threshold is 2. For each $d'<d$ we find the critical exponent for the event that a $d'$-dimensional subtorus becomes open and compute the limiting value of its probability under the critical scaling. For even $d'$, we use the Chen-Stein method to show that the number of $d'$-dimensional subtori that bec...

Consider a graph $G$ and an initial random configuration, where each node is black with probability $p$ and white otherwise, independently. In discrete-time rounds, each node becomes black if it has at least $r$ black neighbors and white otherwise. We prove that this basic process exhibits a threshold behavior with two phase transitions when the underlying graph is a $d$-dimensional torus and identify the threshold values.

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...

We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to determine the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel, and the latter provides an alternative and simpler proof for one of their main results.