#### Title

### The time of bootstrap percolation with dense initial sets

#### Abstract

Let r σ N. 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. When p is close to 1, we study the distribution of the time 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-neighbour bootstrap percolation on the d-dimensional discrete torus Tdn. Moreover, we show that for certain ranges of p = p(n), the time at which percolation occurs is concentrated either on a single value or on two consecutive values. We also prove corresponding results for the modified d-neighbour rule. © Institute of Mathematical Statistics 2014.

#### Publication Title

Annals of Probability

#### Recommended Citation

Bollobás, B., Holmgren, C., Smith, P., & Uzzell, A.
(2014). The time of bootstrap percolation with dense initial sets.* Annals of Probability**, 42* (4), 1337-1373.
https://doi.org/10.1214/12-AOP818