## The Annals of Probability

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

### The sharp threshold for the Duarte model

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

#### Abstract

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

**Source**

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

**Dates**

Received: March 2016

Revised: October 2016

First available in Project Euclid: 12 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1513069259

**Digital Object Identifier**

doi:10.1214/16-AOP1163

**Mathematical Reviews number (MathSciNet)**

MR3737910

**Zentralblatt MATH identifier**

06838119

**Subjects**

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

Secondary: 60C05: Combinatorial probability

**Keywords**

Bootstrap percolation monotone cellular automata Duarte model critical probability sharp threshold

#### Citation

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. https://projecteuclid.org/euclid.aop/1513069259