膨胀-修改系统中的精确标度
Exact scaling in the expansion-modification system
摘要：
这项工作是专门研究的缩放，以及随之而来的幂律行为，相关函数的突变复制模型称为扩展修改系统。后者是一个生物学启发的随机替换模型的基因组进化，这是一个二进制字母表上定义的，并依赖于一个参数解释为\emph{mutation probability}。我们证明了这个系统的时间演化是这样的，任何初始措施收敛到一个独特的固定表现出衰减的相关性不慢于幂律。然后，我们证明，突变概率的一个显着的范围内，衰减的相关性确实遵循幂律与标度指数平滑依赖于突变概率。最后，我们提出了一个参数，它允许我们给出一个封闭的表达式为相应的标度指数的所有值的突变概率。这样的标度指数原来是参数的分段光滑函数。
英文摘要：
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 \emph{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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