Is there a discrete space Markov chain, starting from a fixed state, whose stationary distribution is a multimodal distribution and that mixes in polynomial time?

For example, Ising model on say a complete graph has a multimodal stationary distribution at low temperature. Critical $\beta$ (i.e. inverse temperature) is known to be $\beta_c=1$. So $\beta>1$ is low temperature regime. Single-site Glauber dynamics for Ising model at low temperature on a complete graph is known to mix exponentially slow. Here is a proof. I believe faster mixing Swendsen-Wang dynamics which update multiple spins in one step also mix exponentially slow at low temperature.

So I am wondering if there is a fundamental barrier and no Markov chain can mix in polynomial time, starting from a fixed state, at low temperature for Ising model on say a complete graph? Or is it that no one has found such a Markov chain yet?

Coupling is often used to prove upper bounds on mixing time. If there exists a contractive coupling, then a chain mixes in polynomial time. Here is a paper that proves that there exists a contractive coupling for a Markov chain on a continuous state space and a multimodal stationary distribution. For instance, see example 2.4 in this paper which talks about a Gaussian mixture.

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