Abstract
In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model at inverse temperature \({\beta\geq 0}\) are of the form \({\alpha\mu^{+}_\beta + (1-\alpha)\mu^{-}_\beta}\) , where \({\mu^{+}_\beta}\) and \({\mu^{-}_\beta}\) are the two pure phases and \({0\leq\alpha\leq 1}\) . We present here a new approach to this result, with a number of advantages: (a) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (b) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (c) this new approach might be applicable to systems for which the classical method fails.
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Coquille, L., Velenik, Y. A finite-volume version of Aizenman–Higuchi theorem for the 2d Ising model. Probab. Theory Relat. Fields 153, 25–44 (2012). https://doi.org/10.1007/s00440-011-0339-6
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DOI: https://doi.org/10.1007/s00440-011-0339-6
Keywords
- Ising model
- Gibbs states
- Translation invariance
Mathematics Subject Classification (2000)
- 60K35
- 82B20