Bootstrap percolation is local
arxiv(2024)
摘要
Metastability thresholds lie at the heart of bootstrap percolation theory.
Yet proving precise lower bounds is notoriously hard. We show that for two of
the most classical models, two-neighbour and Froböse, upper bounds are sharp
to essentially arbitrary precision, by linking them to their local
counterparts.
In Froböse bootstrap percolation, iteratively, any vertex of the square
lattice that is the only healthy vertex of a 1×1 square becomes infected
and infections never heal. We prove that if vertices are initially infected
independently with probability p→0, then with high probability the origin
becomes infected after
exp(π^2/6p-π√(2+√(2))/√(p)+O(log^2(1/p))/√(p))
time steps. We achieve this by
proposing a new paradigmatic view on bootstrap percolation based on locality.
Namely, we show that studying the Froböse model is equivalent in an extremely
strong sense to studying its local version. As a result, we completely bypass
Holroyd's classical but technical hierarchy method, yielding the first term
above and systematically used throughout bootstrap percolation for the last two
decades. Instead, the proof features novel links to large deviation theory,
eigenvalue perturbations and others.
We also use the locality viewpoint to resolve the so-called bootstrap
percolation paradox. Indeed, we propose and implement an exact (deterministic)
algorithm which exponentially outperforms previous Monte Carlo approaches. This
allows us to clearly showcase and quantify the slow convergence we prove
rigorously.
The same approach applies, with more extensive computations, to the
two-neighbour model, in which vertices are infected when they have at least two
infected neighbours and do not recover. We expect it to be applicable to a
wider range of models and correspondingly conclude with a number of open
problems.
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