Sharp threshold for two-dimensional majority dynamics percolation

TL;DR

This paper establishes a sharp threshold for percolation in two-dimensional majority dynamics, providing bounds on critical probabilities and extending results to voter model percolation, based on differential inequalities and OSSS inequality.

Contribution

It introduces a sharp threshold phenomenon for 2D majority dynamics percolation and derives bounds on critical probabilities, extending to voter model percolation.

Findings

Sharp threshold at critical probability p_c(t) with polynomial window

Stretched-exponential bounds on one-arm event probability in subcritical phase

Analogous results for voter model percolation

Abstract

In this work we consider the two-dimensional percolation model arising from the majority dynamics process at a given time $t\in\mathbb{R}_+$. We show the emergence of a sharp threshold phenomenon for the box crossing event at the critical probability parameter $p_c(t)$ with polynomial size window. We then use this result in order to obtain stretched-exponential bounds on the one-arm event probability in the subcritical phase. Our results are based on differential inequalities derived from the OSSS inequality, inspired by the recent developments by Ahlberg, Broman, Griffiths, and Morris and by Duminil-Copin, Raoufi, and Tassion. We also provide analogous results for percolation in the voter model.