# The sharp threshold for bootstrap percolation in all dimensions

@article{Balogh2010TheST, title={The sharp threshold for bootstrap percolation in all dimensions}, author={J{\'o}zsef Balogh and B'ela Bollob'as and Hugo Duminil-Copin and Robert Morris}, journal={arXiv: Probability}, year={2010} }

In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the d-dimensional grid $[n]^d$. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine $p_c([n]^d,r…

## 166 Citations

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The main question is to determine the critical probability pc([n]d, r) at which percolation becomes likely, and to give bounds on the size of the critical window.

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By bootstrap percolation we mean the following deterministic process on a graph G. Given a set A of vertices "infected" at time 0, new vertices are subsequently infected, at each time step, if they…

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Abstract. In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V (G) is chosen independently at random, with density p, and new vertices are subsequently…

Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box

- MathematicsElectron. J. Comb.
- 2018

The maximal percolation time for r-neighbour bootstrap percolating on the hypercube for all $r$ as the dimension $d$ goes to infinity up to a logarithmic factor 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).

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- 2018

The main question is to determine pc (n, r, d), the density at which percolation (infection of the entire grid) becomes likely, and to determine the size of the minimal percolating sets in all dimensions and for all values of the infection parameter.

Bootstrap percolation on Galton-Watson trees

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- 2013

Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number $r$, the $r$-neighbour bootstrap process…

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Three-dimensional 2-critical bootstrap percolation: The stable sets approach

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