# 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}
}
• Published 16 October 2010
• Mathematics
• arXiv: Probability
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 $p_c([n]^d,r… 166 Citations ## Figures from this paper Bootstrap Percolation in High Dimensions • Mathematics Combinatorics, Probability and Computing • 2010 The main question is to determine the critical probability pc([n]d, r) at which percolation becomes likely, and to give bounds on the size of the critical window. An Improved Upper Bound for Bootstrap Percolation in All Dimensions It is shown that for all d ⩽ r ⩾ 2 there exists a constant cd,r > 0 such that if n is sufficiently large, then p_c (d, r) is a bound on the second term in the expansion of the critical probability when G = [n]d and d⩾ r 2. Bootstrap percolation in three dimensions • Mathematics • 2009 By bootstrap percolation we mean the following deterministic process on a graph G. Given a set A of vertices "infected" at time 0, new vertices are subsequently infected, at each time step, if they The$d$-dimensional bootstrap percolation models with threshold at least double exponential Consider a p-random subset A of initially infected vertices in the discrete cube [L], and assume that the neighbourhood of each vertex consists of the ai nearest neighbours in the ±ei-directions for PR ] 1 4 Ju l 2 01 0 BOOTSTRAP PERCOLATION IN HIGH DIMENSIONS • Mathematics • 2021 Abstract. In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ V (G) is chosen independently at random, with density p, and new vertices are subsequently Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box The maximal percolation time for r-neighbour bootstrap percolating on the hypercube for all$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$, as established by Przykucki (2012). Line percolation • Mathematics Random Struct. Algorithms • 2018 The main question is to determine pc (n, r, d), the density at which percolation (infection of the entire grid) becomes likely, and to determine the size of the minimal percolating sets in all dimensions and for all values of the infection parameter. Bootstrap percolation on Galton-Watson trees • Mathematics • 2013 Bootstrap percolation is a type of cellular automaton which has been used to model various physical phenomena, such as ferromagnetism. For each natural number$r$, the$r$-neighbour bootstrap process Extremal Bounds for Bootstrap Percolation in the Hypercube • Mathematics Electron. Notes Discret. Math. • 2017 Three-dimensional 2-critical bootstrap percolation: The stable sets approach Consider a p-random subset A of initially infected vertices in the discrete cube [L], and assume that the neighbourhood of each vertex consists of the ai nearest neighbours in the ±ei-directions for ## References SHOWING 1-10 OF 59 REFERENCES Bootstrap Percolation in High Dimensions • Mathematics Combinatorics, Probability and Computing • 2010 The main question is to determine the critical probability pc([n]d, r) at which percolation becomes likely, and to give bounds on the size of the critical window. Bootstrap percolation in three dimensions • Mathematics • 2009 By bootstrap percolation we mean the following deterministic process on a graph G. Given a set A of vertices "infected" at time 0, new vertices are subsequently infected, at each time step, if they Majority Bootstrap Percolation on the Hypercube • Mathematics Combinatorics, Probability and Computing • 2009 The case when the growth of d to ∞ is not excessively slow is studied, and it is shown that the critical probability is 1/2 + o(1) if d ≥ (log log n)2 log log log n, and much stronger bounds in the case that G is the hypercube, [2]d. Majority bootstrap percolation on the random graph G(n,p) • Mathematics • 2012 Majority bootstrap percolation on the random graph$G_{n,p}$is a process of spread of "activation" on a given realisation of the graph with a given number of initially active nodes. At each step Bootstrap Percolation on Infinite Trees and Non-Amenable Groups • Mathematics Combinatorics, Probability and Computing • 2006 It is proved that on any$2k-regular non-amenable graph, the critical probability for the $k$-rule is strictly positive, and that in any rooted tree $T$ there is a way of erasing children of the root, together with all their descendants, and repeating this for all remaining children, and so on.
Bootstrap percolation on the random regular graph
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Random Struct. Algorithms
• 2007
Here, thanks to a “principle of deferred decisions,” the percolation dynamics is described by a surprisingly simple Markov chain, which is replaced by a deterministic dynamical system, and its integrals are used to show—via exponential supermartingales—that thePercolation process undergoes relatively small fluctuations around the deterministic trajectory.
Slow convergence in bootstrap percolation.
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In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2
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Two-dimensional bootstrap percolation is a cellular automaton in which sites become ‘infected’ by contact with two or more already infected nearest neighbours. We consider these dynamics, which can
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• Physics
• 2001
Bootstrap percolation models, or equivalently certain types of cellular automata, exhibit interesting finite-volume effects. These are studied here at a rigorous level. We find that for an initial