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

Published online by Cambridge University Press:  04 June 2010

J.-R. CHAZOTTES ,
J.-M. GAMBAUDO  and
E. UGALDE
J.-R. CHAZOTTES
Affiliation:
Centre de Physique Théorique, École Polytechnique, 91128 Palaiseau Cedex, France (email: [email protected])
J.-M. GAMBAUDO
Affiliation:
Laboratoire J. A. Dieudonné, UMR CNRS 6621, Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
E. UGALDE
Affiliation:
Instituto de Física, Universidad Autónoma de San Luis Potosí, San Luis Potosí SLP 78290, México
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Abstract

Let 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 1109 - 1161
Copyright
Copyright © Cambridge University Press 2010

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