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Exact Scaling in the Expansion-Modification System

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Abstract

This work is devoted to the study of the scaling, and the consequent power-law behavior, of the correlation function in a mutation-replication model known as the expansion-modification system. The latter is a biology inspired random substitution model for the genome evolution, which is defined on a binary alphabet and depends on a parameter interpreted as a mutation probability. We prove that the time-evolution of this system is such that any initial measure converges towards a unique stationary one exhibiting decay of correlations not slower than a power-law. We then prove, for a significant range of mutation probabilities, that the decay of correlations indeed follows a power-law with scaling exponent smoothly depending on the mutation probability. Finally we put forward an argument which allows us to give a closed expression for the corresponding scaling exponent for all the values of the mutation probability. Such a scaling exponent turns out to be a piecewise smooth function of the parameter.

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Notes

  1. According to our numerical computations.

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Authors and Affiliations

  1. Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Avenida Universidad 1001, Colonia Chamilpa, Cuernavaca C.P. 62209, Morelos, Mexico

    R. Salgado-García

  2. Instituto de Física, Universidad Autónoma de San Luis Potosí, Avenida Manuel Nava 6, Zona Universitaria, 78290, San Luis Potosí, Mexico

    E. Ugalde

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  1. R. Salgado-García
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  2. E. Ugalde
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Correspondence to E. Ugalde.

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Salgado-García, R., Ugalde, E. Exact Scaling in the Expansion-Modification System. J Stat Phys 153, 842–863 (2013). https://doi.org/10.1007/s10955-013-0866-x

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