Bootstrap percolation on the hypercube

Source abstract

In this paper we obtain the critical probability $p_c(Q_{n},r)$ for bootstrap percolation with the infection threshold $r=f(n)n^a$ on the $n$-dimensional hypercube $Q_{n}$, where $0<a<1$ and $f(n)=n^{o(1)}$. More precisely, we obtain the exact first-order and second-order terms of $p_c(Q_{n},r)$ by extending the main theorem of Balogh, Bollob{á}s, and Morris (2009).