1. INTRODUCTION 7

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.