  We study the autocorrelation time of the size of the cluster at the origin in
discrete-time dynamical percolation. We focus on binary trees and
high-dimensional tori, and show in both cases that this autocorrelation time is
linear in the volume in the subcritical regime, but strictly sublinear in the
volume at criticality. This establishes rigorously that the cluster size at the
origin in these models exhibits critical speeding-up. The proofs involve
controlling relevant Fourier coefficients. In the case of binary trees, these
Fourier coefficients are studied explicitly, while for high-dimensional tori we
employ a randomised algorithm argument introduced by Schramm and Steif in the
context of noise sensitivity.
