Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials
2010 ◽
Vol 31
(4)
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pp. 1109-1161
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Keyword(s):
AbstractLet A be a finite set and let ϕ:Aℤ→ℝ be a locally constant potential. For each β>0 (‘inverse temperature’), there is a unique Gibbs measure μβϕ. We prove that as β→+∞, the family (μβϕ)β>0 converges (in the weak-* topology) to a measure that we characterize. This measure is concentrated on a certain subshift of finite type, which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron–Frobenius theorem for matrices à la Birkhoff. The crucial idea we bring is a ‘renormalization’ procedure which explains convergence and provides a recursive algorithm for computing the weights of the ergodic decomposition of the limit.
1990 ◽
Vol 10
(3)
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pp. 421-449
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Keyword(s):
2017 ◽
Vol 28
(03)
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pp. 263-287
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1983 ◽
Vol 3
(4)
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pp. 541-557
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Keyword(s):
2008 ◽
Vol 28
(1)
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pp. 167-209
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1994 ◽
Vol 14
(2)
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pp. 213-235
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1989 ◽
Vol 9
(3)
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pp. 561-570
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