Before revision:  This  provides a nearly optimal bound (up to the \log\log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no previous bound was known.
After revision:  This provides a nearly optimal bound (up to the \log\log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no non-trivial bound was previously known.