Abstract
Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $\theta $ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor is a cycle. We identify the asymptotic behavior of the critical probability and show that, when $\theta $ is odd, there are two qualitatively distinct phases: the transition from low to high probability of spanning as the initial density increases is sharp or gradual, depending on the size of $m$.
Highlights
Given a graph G = (V, E), bootstrap percolation with threshold θ is a discrete-time growth process that, starting from an initial configuration ω ∈ {0, 1}V, generates an increasing sequence of configurations ω = ω0, ω1
The most natural object of study is the event Span = {ω∞ ≡ 1} that spanning occurs
Some work has been done on the Hamming torus Knd, which, as the Cartesian product of d complete graphs of n vertices, has the same vertex set as the lattice cube, but a much larger set of edges, which makes many percolation questions fundamentally different [Siv, GHPS, Sli]
Summary
Some work has been done on the Hamming torus Knd, which, as the Cartesian product of d complete graphs of n vertices, has the same vertex set as the lattice cube, but a much larger set of edges, which makes many percolation questions fundamentally different [Siv, GHPS, Sli]. 0 a ∈ (0, 1) Papc (Span) → 1 a ∈ (1, ∞) Sharp transitions results have been proved in remarkable generality [FK]. They hold for bootstrap percolation on the lattices Zdn, where much more is proved [Hol, BB, BBM, BBDM]. Denote by log(k) the kth iterate of log and let λ(d, θ) be the bootstrap percolation scaling constant for the lattice Zdm defined in [BBDM].
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