The sharp threshold for the Duarte model

TL;DR

This paper establishes a precise percolation threshold for the Duarte bootstrap model, a significant step in understanding critical phenomena in models with drift, resolving a longstanding open problem.

Contribution

The paper refines previous techniques to determine a sharp percolation threshold specifically for the Duarte model, addressing a question posed in 1995.

Findings

Determined the sharp percolation threshold for the Duarte model.

Resolved a longstanding open problem from 1995.

Developed refined techniques for bootstrap percolation analysis.

Abstract

The class of critical bootstrap percolation models in two dimensions was recently introduced by Bollob\'as, 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.