Maximal Percolation Time in Hypercubes Under 2-Bootstrap Percolation

Bootstrap percolation is one of the simplest cellular automata. In r-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 r already infected neighbours become infected. Percolation occurs if eventually every vertex is infected. In this paper we prove that in the case of 2-bootstrap percolation on the n-dimensional hypercube the maximal time the process can take to eventually infect the entire vertex set is bn2 3 c.