Slow mixing of Glauber dynamics via topological obstructions

Many local Markov chains based on Glauber dynamics are known to undergo a phase transition as a parameter λ of the system is varied. For independent sets on the 2-dimensional Cartesian lattice, the Gibbs distribution assigns each independent set a weight λ|I|, and the Markov chain adds or deletes a single vertex at each step. It is believed that there is a critical point λc ≈ 3.79 such that for λ < λc, local dynamics converge in polynomial time while for λ> λc they require exponential time. We introduce a new method for showing slow mixing based on the presence or absence of certain topological obstructions in the independent sets. Using elementary arguments, we show that Glauber dynamics will be slow for sampling independent sets in 2 dimensions when λ ≥ 56.812, improving on the best known bound while using simpler arguments. We also show they are slow on the torus when λ ≥ 50.5253.