Abstract
For the hard-core model (independent sets) on ℤ2 with fugacity λ, we give the first explicit result for phase coexistence by showing that there are multiple Gibbs states for all λ > 5.3646. Our proof begins along the lines of the standard Peierls argument, but we add two significant innovations. First, building on the idea of fault lines introduced by Randall [19], we construct an event that distinguishes two boundary conditions and yet always has long contours associated with it, obviating the need to accurately enumerate short contours. Second, we obtain vastly improved bounds on the number of contours by relating them to a new class of self-avoiding walks on an oriented version of ℤ2. We also extend our characterization of fault lines to show that local Markov chains will mix slowly when λ > 5.3646 on lattice regions with periodic (toroidal) boundary conditions and when λ > 7.1031 with non-periodic (free) boundary conditions. The arguments here rely on a careful analysis that relates contours to taxi walks and represent a sevenfold improvement to the previously best known values of λ [19].
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Blanca, A., Galvin, D., Randall, D., Tetali, P. (2013). Phase Coexistence and Slow Mixing for the Hard-Core Model on ℤ2 . In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2013 2013. Lecture Notes in Computer Science, vol 8096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40328-6_27
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DOI: https://doi.org/10.1007/978-3-642-40328-6_27
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