Abstract
We study a basic model for mutations. We derive exact formulae for the mean time needed to discover the master sequence, the mean returning time to the initial state, or to any Hamming class. These last two formulae are the same than the formulae obtained by Mark Kac for the Ehrenfest model.
Highlights
According to the Darwinian paradigm, the evolution of living creatures is driven by two main forces: mutations and selection
Mutations create new forms of behaviour or new characters, some less fit to their environment, some more, whereas selection favours the reproduction of fitter individuals
Selection alone would result in uniform populations, lacking in genetic diversity
Summary
According to the Darwinian paradigm, the evolution of living creatures is driven by two main forces: mutations and selection. Single individual, we follow his lineage and we compute the time needed for the genotype 0 · · · 0 to be discovered. The formula of Theorem 1.2 is a straightforward consequence of the classical result expressing the invariant probability measure of a Markov chain in terms of mean recurrence times.
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