Slow Mixing of Markov Chains Using Fault Lines and Fat Contours

We show that local dynamics require exponential time for two sampling problems: independent sets on the triangular lattice (the hard-core lattice gas model) and weighted even orientations of the Cartesian lattice (the 8-vertex model). For each problem, there is a parameter i¾?known as the fugacity such that local Markov chains are expected to be fast when i¾?is small and slow when i¾?is large. However, establishing slow mixing for these models has been a challenge because standard contour arguments typically used to show that a chain has small conductance do not seem sufficient. We modify this approach by introducing the notion of fat contoursthat can have nontrivial d-dimensional volume and use these to establish slow mixing of local chains defined for these models.

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