  We prove that the spectral gap of the Swendsen-Wang dynamics for the
random-cluster model on arbitrary graphs with m edges is bounded above by 16 m
log m times the spectral gap of the single-bond (or heat-bath) dynamics. This
and the corresponding lower bound imply that rapid mixing of these two dynamics
is equivalent.
  Using the known lower bound on the spectral gap of the Swendsen-Wang dynamics
for the two dimensional square lattice $Z_L^2$ of side length L at high
temperatures and a result for the single-bond dynamics on dual graphs, we
obtain rapid mixing of both dynamics on $\Z_L^2$ at all non-critical
temperatures. In particular this implies, as far as we know, the first proof of
rapid mixing of a classical Markov chain for the Ising model on $\Z_L^2$ at all
temperatures.
