Critical speeding-up in dynamical percolation

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.