Mathematics – Combinatorics
Scientific paper
2012-04-14
Mathematics
Combinatorics
19 pages, 2 figures
Scientific paper
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 interesting to determine 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 second term in the expansion of the critical probability when $G = [n]^d$ and $d \geq r \geq 2$. We show that there exists a constant $c > 0$ such that for all $d \geq r \geq 2$, \[p_c([n]^d, r) \leq (\dfrac{\lambda(d,r)}{\log_{(r-1)}(n)} - \dfrac{c}{(\log_{(r-1)}(n))^{2 - 1/(2d - 2)}})^{d-r+1}\] as $n \to \infty$, where $\lambda(d,r)$ is an exact constant and $\log_{(k)}(n)$ denotes the $k$-times iterated logarithm of $n$.
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