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Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials

  • J.-R. CHAZOTTES (a1), J.-M. GAMBAUDO (a2) and E. UGALDE (a3)

Let A be a finite set and let ϕ:A→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.

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[8] H.-O. Georgii . Gibbs Measures and Phase Transitions (de Gruyter Studies in Mathematics, 9). Walter de Gruyter & Co., Berlin, 1988.

[12] D. Ruelle . Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edn.(Cambridge Mathematical Library). Cambridge University Press, Cambridge, 2004.

[13] E. Seneta . Non-negative Matrices and Markov Chains (Springer Series in Statistics). Springer, Berlin, 1981.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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