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Subcritical bootstrap percolation via Toom contours. (English) Zbl 1515.60313

Summary: In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions, but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom’s North-East-Center majority rule cellular automaton.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60C05 Combinatorial probability
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
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