on the population size and genome length has been investigated numerically by
Elena, Wilke, Ofria and Lenski [16]. Here we show rigorously the existence of a
critical population size for the sharp peak landscape in a specific asymptotic regime.
The existence of a critical population size for the emergence of a quasispecies is a
pleasing result: it shows that, even under the action of selection forces, a form of
cooperation is necessary to create a quasispecies. Moreover the critical population
size is much smaller than the cardinality of the possible genotypes. In conclusion,
even in the very simple framework of the Moran model on the sharp peak landscape,
cooperation is necessary to achieve the survival of the master sequence.
As emphasized by Eigen in [14], the error threshold phenomenon is similar
to a phase transition in statistical mechanics. Leuth¨ ausser established a formal
correspondence between Eigen’s model and an anisotropic Ising model [25]. Sev-
eral researchers have employed tools from statistical mechanics to analyze models
of biological evolution, and more specifically the error threshold: see the nice re-
view written by Baake and Gabriel [3]. Baake investigated the so–called Onsager
landscape in [4]. This way she could transfer to a biological model the famous com-
putation of Onsager for the two dimensional Ising model. Saakian, Deem and Hu
[34] compute the variance of the mean fitness in a finite population model in order
to control how it approximates the infinite population model. Deem, Mu˜ noz and
Park [33] use a field theoretic representation in order to derive analytical results.
We were also very much inspired by ideas from statistical mechanics, but with
a different flavor. We do not use exact computations, rather we rely on softer
tools, namely coupling techniques and correlation inequalities. These are the basic
tools to prove the existence of a phase transition in classical models, like the Ising
model or percolation. We seek large deviation estimates rather than precise scaling
relations in our asymptotic regime. Of course the outcome of these techniques is
very rough compared to exact computations, yet they are much more robust and
their range of applicability is much wider. The model is presented in the next
chapter and the main results in chapter 3. The remaining chapters are devoted to
the proofs. In the appendix we recall several classical results of the theory of finite
Markov chains.
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