  Glauber dynamics of the Ising model on a random regular graph is known to mix
fast below the tree uniqueness threshold and exponentially slowly above it. We
show that Kawasaki dynamics of the canonical ferromagnetic Ising model on a
random $d$-regular graph mixes fast beyond the tree uniqueness threshold when
$d$ is large enough (and conjecture that it mixes fast up to the tree
reconstruction threshold for all $d\geq 3$). This result follows from a more
general spectral condition for (modified) log-Sobolev inequalities for
conservative dynamics of Ising models. The proof of this condition in fact
extends to perturbations of distributions with log-concave generating
polynomial.
