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Proving slow mixing with fault lines and fat contours

Presented by: 
D Randall [Sao Paulo State]
Thursday 27th March 2008 - 14:00 to 14:30
INI Seminar Room 1

We show that for several sampling problems, local dynamics require exponential time to converge to equilibrium. In particular, we apply our techniques to the problems of sampling independent sets on the triangular lattice (the hard-core lattice gas model) and the weighted even orientations of the Cartesian lattice (the 8-vertex model). For each problem, there is a single parameter (corresponding to temperature or fugacity) such that local Markov chains are expected to be fast at high temperature and slow at low termpature. However, establishing slow mixing for these models has been a challenge because standard Peierls arguments based on (d-1)-dimensional contours do not seem to work. Here we extend this approach to "fat contours" that have nontrivial d-dimensional volume, and use these to establish slow mixing of the local chains. In addition, we show that restricting the contours to fat contours containing "fault lines" allow us to further simplify these proofs.

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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons