Abstract
In $r$-neighbour bootstrap percolation, vertices (sites) of a graph $G$ become "infected" in each round of the process if they have $r$ neighbours already infected. Once infected, they remain such. An initial set of infected sites is said to percolate if every site is eventually infected. We determine the maximal percolation time for $r$-neighbour bootstrap percolation on the hypercube for all $r \geq 3$ as the dimension $d$ goes to infinity up to a logarithmic factor. Surprisingly, it turns out to be $\frac{2^d}{d}$, which is in great contrast with the value for $r=2$, which is quadratic in $d$, as established by Przykucki (2012). Furthermore, we discover a link between this problem and a generalisation of the well-known Snake-in-the-Box problem.
Highlights
Bootstrap percolation was introduced in 1979 by Chalupa, Leath and Reich [11] as a simplified monotone version of ferromagnetic dynamics and it is in particular related to Glauber dynamics of the Ising model
We discover a link between this problem and a generalisation of the well-known Snake-in-the-Box problem
We show that the maximal percolation time goes from close to the trivial lower bound 1, as found in [24], to close to the trivial upper bound 2d
Summary
Bootstrap percolation was introduced in 1979 by Chalupa, Leath and Reich [11] as a simplified monotone version of ferromagnetic dynamics and it is in particular related to Glauber dynamics of the Ising model. An essential ingredient for this bound is a new link we establish between bootstrap percolation and the very well-known snakein-the-box problem, which concerns long induced paths and cycles in the hypercube. It was introduced by Kautz in the late 50s [19] and has a wide range of applications, namely in coding, error-correction and others. It was first proved in [12, 15] that the maximal length of a snake-in-the-box is 2d up to a constant factor, though its correct asymptotic value is not yet known.
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