Abstract
We study the threshold θ bootstrap percolation model on the homogeneous tree with degree b+1, 2≤θ≤b, and initial density p. It is known that there exists a nontrivial critical value for p, which we call p f , such that a) for p>p f , the final bootstrapped configuration is fully occupied for almost every initial configuration, and b) if p<p f , then for almost every initial configuration, the final bootstrapped configuration has density of occupied vertices less than 1. In this paper, we establish the existence of a distinct critical value for p, p c , such that 0<p c <p f , with the following properties: 1) if p≤p c , then for almost every initial configuration there is no infinite cluster of occupied vertices in the final bootstrapped configuration; 2) if p>p c , then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration. Moreover, we show that 3) for p<p c , the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail; 4) at p=p c , the expected occupied cluster size in the final bootstrapped configuration is infinite; 5) the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on [0,p f ] and analytic on (p c ,p f ), admitting an analytic continuation from the right at p c and, only in the case θ=b, also from the left at p f .
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L.R.G. Fontes partially supported by the Brazilians CNPq through grants 475833/2003-1, 307978/2004-4 and 484351/2006-0, and FAPESP through grant 04/07276-2.
R.H. Schonmann partially supported by the American N.S.F. through grant DMS-0300672.
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Fontes, L.R.G., Schonmann, R.H. Bootstrap Percolation on Homogeneous Trees Has 2 Phase Transitions. J Stat Phys 132, 839–861 (2008). https://doi.org/10.1007/s10955-008-9583-2
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DOI: https://doi.org/10.1007/s10955-008-9583-2