Journal of Statistical Physics

, Volume 171, Issue 2, pp 269–287 | Cite as

Constant-Length Random Substitutions and Gibbs Measures

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Abstract

This work is devoted to the study of processes generated by random substitutions over a finite alphabet. We prove, under mild conditions on the substitution’s rule, the existence of a unique process which remains invariant under the substitution, and which exhibits a polynomial decay of correlations. For constant-length substitutions, we go further by proving that the invariant state is precisely a Gibbs measure which can be obtained as the projective limit of its natural Markovian approximations. We end up the paper by studying a class of substitutions whose invariant state is the unique Gibbs measure for a hierarchical two-body interaction.

Keywords

Gibbs measures Random substitutions Projective convergence 

Notes

Acknowledgements

We thank CONACyT-México and Fundación Marcos Moshinsky for their financial support through Grant CB-2014-237324-F and through “Cátedra Marcos Moshinsky 2016” respectively. The final stage of the work was done during a visit of E.U. on CPhT-École Polytechnique, during which he benefited form the financial support of École Polytechnique, and the hospitality and scientific advise of Prof. J.-R. Chazottes. C.M. thanks the Instituto de Física UASLP for the warm hospitality during a one-month visit at the early stage of this work which was partially supported by the CONICYT-FONDECYT Postdoctoral Grant No. 3140572.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto de FísicaUniversidad Autónoma de San Luis PotosíSan Luis Potosí Mexico
  2. 2.División de Matemáticas AplicadasInstituto Potosino de Investigación Científica y TecnológicaSan Luis PotosíMexico

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