Abstract
We consider an Ising spin system with Kac potentials in a torus of ℤd,d>-2, and fix the temperature below its Lebowitz-Penrose critical value. We prove that when the Kac scaling parameter γ vanishes, the log of the probability of an interface becomes proportional to its area and the surface tension, related to the proportionality constant, converges to the van der Waals surface tension. The results are based on the analysis of the rate functionals for Gibbsian large deviations and on the proof that they Γ-converge to the perimeter functional of geometric measure theory (which extends the notion of area). Our considerations include nonsmooth interfaces, proving that the Gibbsian probability of an interface depends only on its area and not on its regularity.
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Alberti, G., Bellettini, G., Cassandro, M. et al. Surface tension in Ising systems with Kac potentials. J Stat Phys 82, 743–796 (1996). https://doi.org/10.1007/BF02179792
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DOI: https://doi.org/10.1007/BF02179792