Abstract
We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least θ occupied neighbors, occupied sites remain occupied forever. It is known that, when b>θ≥2, the limiting density q=q(p) of occupied sites exhibits a jump at some p T=p T(b,θ)∈(0,1) from q T:=q(p T)<1 to q(p)=1 when p>p T. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p T+h with h>0 and show that, as h ↓0, the system lingers around the “critical” state for time order h −1/2 and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q∈(q T,1) converges, as h ↓0, to a well-defined measure.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Biskup, M., Schonmann, R.H. Metastable Behavior for Bootstrap Percolation on Regular Trees. J Stat Phys 136, 667–676 (2009). https://doi.org/10.1007/s10955-009-9798-x
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DOI: https://doi.org/10.1007/s10955-009-9798-x