Kawasaki dynamics mixing conjecture for the fixed-magnetization Ising model

Let $G$ be a graph on $n$ vertices with maximum degree $\Delta$, and consider the fixed-magnetization Ising model with inverse temperature $\beta$ and magnetization parameter $\eta$. The Kawasaki dynamics has two versions: local flip dynamics, proposing swaps of spins across an edge, and global flip dynamics, proposing swaps of arbitrary spins. Let $\beta_c(\Delta)$ and $\eta_c(\Delta,\beta)$ denote the threshold parameters in the paper.

Kawasaki dynamics mixing conjecture. For $\beta<\beta_c(\Delta)$, the Kawasaki dynamics mix in time polynomial in $n$ for any fixed magnetization and any graph $G$ of maximum degree $\Delta$. For $\beta>\beta_c(\Delta)$ and $|\eta|>\eta_c(\Delta,\beta)$, they mix in time polynomial in $n$ for any fixed magnetization $k\geq\eta n$ and any such graph. In both cases, the global flip dynamics mix in time $O(n\log n)$.

This conjecture concerns the efficiency of a standard local Markov-chain sampler for the fixed-magnetization Ising model, complementing the paper’s approximation results. The source does not provide evidence of a resolution, so the conjecture remains open; the relation between the one-sentence rapid-mixing claim and the more detailed parameterized formulation should be checked.