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Subcritical $ \mathcal{U}$-bootstrap percolation models have non-trivial phase transitions


Authors: Paul Balister, Béla Bollobás, Michał Przykucki and Paul Smith
Journal: Trans. Amer. Math. Soc. 368 (2016), 7385-7411
MSC (2010): Primary 60K35, 82B26, 60C05
Published electronically: January 27, 2016
MathSciNet review: 3471095
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Abstract: We prove that there exist natural generalizations of the classical bootstrap percolation model on $ \mathbb{Z}^2$ that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property.

Van Enter (1987) (in the case $ d=r=2$) and Schonmann (1992) (for all $ d \geq r \geq 2$) proved that $ r$-neighbour bootstrap percolation models have trivial critical probabilities on $ \mathbb{Z}^d$ for every choice of the parameters $ d \geq r \geq 2$: that is, an initial set of density $ p$ almost surely percolates $ \mathbb{Z}^d$ for every $ p>0$. These results effectively ended the study of bootstrap percolation on infinite lattices.

Recently Bollobás, Smith and Uzzell introduced a broad class of percolation models called $ \mathcal {U}$-bootstrap percolation, which includes $ r$-neighbour bootstrap percolation as a special case. They divided two-dimensional $ \mathcal {U}$-bootstrap percolation models into three classes - subcritical, critical and supercritical - and they proved that, like classical 2-neighbour bootstrap percolation, critical and supercritical $ \mathcal {U}$-bootstrap percolation models have trivial critical probabilities on $ \mathbb{Z}^2$. They left open the question as to what happens in the case of subcritical families. In this paper we answer that question: we show that every subcritical $ \mathcal {U}$-bootstrap percolation model has a non-trivial critical probability on $ \mathbb{Z}^2$. This is new except for a certain `degenerate' subclass of symmetric models that can be coupled from below with oriented site percolation. Our results re-open the study of critical probabilities in bootstrap percolation on infinite lattices, and they allow one to ask many questions of subcritical bootstrap percolation models that are typically asked of site or bond percolation.


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Additional Information

Paul Balister
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
Email: pbalistr@memphis.edu

Béla Bollobás
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom – and – Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152 – and – London Institute for Mathematical Sciences, 35a South Street, Mayfair, London W1K 2XF, United Kingdom
Email: b.bollobas@dpmms.cam.ac.uk

Michał Przykucki
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom – and – London Institute for Mathematical Sciences, 35a South Street, Mayfair, London W1K 2XF, United Kingdom
Email: mp@lims.ac.uk

Paul Smith
Affiliation: IMPA, 110 Estrada Dona Castorina, Jardim Botânico, Rio de Janeiro, 22460-320, Brazil
Email: psmith@impa.br

DOI: https://doi.org/10.1090/tran/6586
Keywords: Bootstrap percolation, phase transitions
Received by editor(s): November 22, 2013
Received by editor(s) in revised form: September 8, 2014, and October 6, 2014
Published electronically: January 27, 2016
Additional Notes: The first author was partially supported by NSF grant DMS 1301614. The second author was partially supported by NSF grant DMS 1301614 and MULTIPLEX no. 317532. The third author was supported by MULTIPLEX no. 317532. The fourth author was supported by a CNPq bolsa PDJ
Article copyright: © Copyright 2016 American Mathematical Society