Majority Bootstrap Percolation on the Hypercube

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 ⊂ V (G), are normally chosen independently at random, each with probability p, say. This process has been extensively studied on the sequence of torus graphs [n], for n = 1, 2, . . ., where d = d(n) is either fixed or a very slowly growing function of n. For example, Cerf and Manzo [14] showed that the critical probability is o(1) if d(n) 6 log∗ n, i.e., if p = p(n) is bounded away from zero then the probability of percolation on [n] tends to one as n →∞. In this paper we study the case when the growth of d to∞ is not excessively slow; in particular, we show that the critical probability is 1/2+o(1) if d > (log log n) log log log n, and give much stronger bounds in the case that G is the hypercube, [2].