The Annals of Probability

The sharp threshold for the Duarte model

Béla Bollobás, Hugo Duminil-Copin, Robert Morris, and Paul Smith

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The class of critical bootstrap percolation models in two dimensions was recently introduced by Bollobás, Smith and Uzzell, and the critical threshold for percolation was determined up to a constant factor for all such models by the authors of this paper. Here, we develop and refine the techniques introduced in that paper in order to determine a sharp threshold for the Duarte model. This resolves a question of Mountford from 1995, and is the first result of its type for a model with drift.

Article information

Ann. Probab. Volume 45, Number 6B (2017), 4222-4272.

Received: March 2016
Revised: October 2016
First available in Project Euclid: 12 December 2017

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Digital Object Identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability

Bootstrap percolation monotone cellular automata Duarte model critical probability sharp threshold


Bollobás, Béla; Duminil-Copin, Hugo; Morris, Robert; Smith, Paul. The sharp threshold for the Duarte model. Ann. Probab. 45 (2017), no. 6B, 4222--4272. doi:10.1214/16-AOP1163.

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