Exact scaling in the expansion-modification system

TL;DR

This paper analyzes the expansion-modification system, a biological mutation model, proving that its correlation decay follows a mutation-dependent power-law with a closed-form scaling exponent.

Contribution

It establishes the power-law decay of correlations in the system and derives a closed-form expression for the scaling exponent across mutation probabilities.

Findings

Correlation functions decay as a power-law with mutation-dependent exponents.

The scaling exponent is a piecewise smooth function of mutation probability.

Convergence towards a unique stationary measure with decay of correlations.

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 \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.