Exact Scaling in the ExpansionModification System
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This work is devoted to the study of the scaling, and the consequent powerlaw behavior, of the correlation function in a mutationreplication model known as the expansionmodification 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 timeevolution of this system is such that any initial measure converges towards a unique stationary one exhibiting decay of correlations not slower than a powerlaw. We then prove, for a significant range of mutation probabilities, that the decay of correlations indeed follows a powerlaw 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. © 2013 Springer Science%2bBusiness Media New York.
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Genome evolution; Powerlaw decay of correlations; Random substitutions
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