The sharp threshold for bootstrap percolation in all dimensions

By Jozsef Balogh, Béla Bollobás, H. Duminil-Copin, Robert Morris. 2010

210 citations recorded by the source corpus.

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 $p_c([n]^d,r)$, the density at which percolation (infection of the entire vertex set) becomes likely. In this paper we prove, for every pair $d \ge r \ge 2$, that there is a constant L(d,r) such that $p_c([n]^d,r) = [(L(d,r) + o(1)) / log_(r-1) (n)]^{d-r+1}$ as $n \to \infty$, where $log_r$ denotes an r-times iterated logarithm. We thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. Moreover, we determine L(d,r) for every pair (d,r).

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4189d6ce821b14f541510ce3ddc444f8e0f2a311

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Majority bootstrap percolation on the random graph G(n,p) (2010, background): “[6] derived a sharp asymptotic for the critical density (i.”
Recent advances in percolation theory and its applications (2015, background): “This has been recently [228] generalized to the finite lattice Z of linear size L as follows”