Abstract
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 λ known as the fugacity such that local Markov chains are expected to be fast when λ is small and slow when λ 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 contours that can have nontrivial d-dimensional volume and use these to establish slow mixing of local chains defined for these models.
Supported in part by NSF grants CCR-0515105 and DMS-0505505.
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Greenberg, S., Randall, D. (2007). Slow Mixing of Markov Chains Using Fault Lines and Fat Contours. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_39
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DOI: https://doi.org/10.1007/978-3-540-74208-1_39
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