An Improved Upper Bound for Bootstrap Percolation in All Dimensions

In $r$-neighbor 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 neighbors. When the elements of $A$ are chosen independently with some probability $p$, it is natural to study the critical probability $p_c(G,r)$ at which it becomes likely that all of $V(G)$ will eventually become infected. Improving a result of Balogh, Bollob\'as, and Morris, we give a bound on the second term in the expansion of the critical probability when $G = [n]^d$ and $d \geq r \geq 2$. We show that for all $d \geq r \geq 2$ there exists a constant $c_{d,r} > 0$ such that if $n$ is sufficiently large, then $$p_c([n]^d, r) \leq \Biggl(\dfrac{\lambda(d,r)}{\log_{(r-1)}(n)} - \dfrac{c_{d,r}}{\bigl(\log_{(r-1)}(n)\bigr)^{3/2}}\Biggr)^{d-r+1},$$ where $\lambda(d,r)$ is an exact constant and $\log_{(k)}(n)$ denotes the $k$-times iterated natural logarithm of $n$.