Journal of Statistical Physics

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

Constant-Length Random Substitutions and Gibbs Measures

  • C. Maldonado
  • L. Trejo-Valencia
  • E. Ugalde


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.


Gibbs measures Random substitutions Projective convergence 



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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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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