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.

翻译：众所周知，在随机正则图上，伊辛模型的Glauber动力学在树唯一性阈值以下快速混合，而在该阈值以上则呈指数缓慢混合。我们证明，当$d$足够大时，随机$d$-正则图上正则铁磁伊辛模型的Kawasaki动力学在树唯一性阈值之外仍能快速混合（并猜想对于所有$d\geq 3$，其混合速度可快至树重构阈值）。该结果源自一类适用于伊辛模型保守动力学的（修正）对数Sobolev不等式的更一般谱条件。事实上，该条件的证明可推广至具有对数凹生成多项式分布的扰动情形。