Percolation phase transition on planar spin systems

TL;DR

This paper proves that the phase transition in certain planar spin system percolation models, including the Ising model and bootstrap processes, is continuous and sharp, with quantitative connectivity estimates.

Contribution

It establishes the sharpness and continuity of phase transitions for dependent percolation models on planar spin systems, with new quantitative estimates.

Findings

Phase transition is continuous and sharp in studied models.

Quantitative estimates on two-point connectivity.

Techniques applicable to various dependent percolation models.

Abstract

In this article we study the sharpness of the phase transition for percolation models defined on top of planar spin systems. The two examples that we treat in detail concern the Glauber dynamics for the Ising model and a Dynamic Bootstrap process. For both of these models we prove that their phase transition is continuous and sharp, providing also quantitative estimates on the two point connectivity. The techniques that we develop in this work can be applied to a variety of different dependent percolation models and we discuss some of the problems that can be tackled in a similar fashion. In the last section of the paper we present a long list of open problems that would require new ideas to be attacked.