An Improved Upper Bound for Bootstrap Percolation in All Dimensions

Abstract

AbstractIn r-neighbour bootstrap percolation on the vertex set of a graph G, a set A of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least r previously infected neighbours. When the elements of A are chosen independently with some probability p, it is natural to study the critical probability pc(G, r) at which it becomes likely that all of V(G) will eventually become infected. Improving a result of Balogh, Bollobás and Morris, we give a bound on the second term in the expansion of the critical probability when G = [n]d and d ⩾ r ⩾ 2. We show that for all d ⩾ r ⩾ 2 there exists a constant cd,r > 0 such that if n is sufficiently large, then $$p_c (\left[ n \right]^d ,{\rm{ }}r){\rm{\le }}\left( {\frac{{\lambda (d,r)}}{{\log _{(r - 1)} (n)}} - \frac{{c_{d,r} }}{{(\log _{(r - 1)} (n))^{3/2} }}} \right)^{d - r + 1} ,$$where λ(d, r) is an exact constant and log(k) (n) denotes the k-times iterated natural logarithm of n.