  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.
