Markov chains for the hard-core model via polymer models
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Jenssen, Keevash and Perkins give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs. Their method is based on using the cluster expansion to obtain a zero-free region for a so-called polymer model partition function in the complex plane, and then approximating the polymer partition function using the Taylor series truncation method of Barvinok. We circumvent the zero-free analysis and the generalisation to complex fugacities, showing that the polymer partition function can be approximated using Glauber dynamics on polymers. The proof that polymer Glauber dynamics mixes rapidly is easy and is based on using the sizes of the disagreeing polymers as a distance function. We then use Markov chain comparison to show that the hard-core partition function can be approximated using spin Glauber dynamics restricted to even and odd dominant portions of the state space. Our techniques also apply to the Ferromagnetic Potts model.
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