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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)
Abstract
Abstract

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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[1]Birkhoff G.. Projective contraction of a cone equipped with its associated Hilbert metric. Trans. Amer. Math. Soc. 85 (1957), 219227.
[2]Bowen R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, 2nd edn.(Lecture Notes in Mathematics, 470). Springer, Berlin, 2008.
[3]Brémont J.. Gibbs measures at temperature zero. Nonlinearity 16 (2003), 419426.
[4]Chazottes J.-R. and Hochman M.. On the zero-temperature limit of Gibbs states. Comm. Math. Phys. (2010), to appear, http://www.springerlink.com/index/1083734146562t7g.pdf.
[5]Chazottes J.-R., Ramírez L. A. and Ugalde E.. Finite type approximations to Gibbs measures on sofic subshifts. Nonlinearity 18(1) (2005), 445465.
[6]Conze J.-P. and Guivarc’h Y.. Croissance des sommes ergodiques et principe variationnel. Unpublished manuscript, 1995.
[7]van Enter A. C. D. and Ruszel W. M.. Chaotic temperature dependence at zero temperature. J. Stat. Phys. 127 (2007), 567573.
[8]Georgii H.-O.. Gibbs Measures and Phase Transitions (de Gruyter Studies in Mathematics, 9). Walter de Gruyter & Co., Berlin, 1988.
[9]Jenkinson O.. Geometric barycentres of invariant measures for circle maps. Ergod. Th. & Dynam. Sys. 21(2) (2001), 511532.
[10]Leplaideur R.. A dynamical proof for the convergence of Gibbs measures at temperature zero. Nonlinearity 18 (2005), 28472880.
[11]Nekhoroshev N. N.. Asymptotics of Gibbs measures in one-dimensional lattice models. Moscow Univ. Math. Bull. 59(1) (2004), 1015.
[12]Ruelle D.. Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edn.(Cambridge Mathematical Library). Cambridge University Press, Cambridge, 2004.
[13]Seneta E.. Non-negative Matrices and Markov Chains (Springer Series in Statistics). Springer, Berlin, 1981.
[14]van Enter A. C. D., Fernández R. and Sokal A.. Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72(5–6) (1993), 8791165.
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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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