Exact solution of the quasispecies model in a sharply peaked fitness landscape

We reconsider Eigen's quasispecies model for competing self-reproductive macromolecules in populations characterized by a single-peaked fitness landscape. The use of ideas and tools borrowed from polymer theory and statistical mechanics allows us to exactly solve the model for generic DNA lengths d. The mathematical shape of the quasispecies confined around the master sequence is perturbatively found in powers of lid at large d. We rigorously prove the existence of the error-threshold phenomena and study the quasispecies formation in the general context of critical phase transitions in physics. No sharp transitions exist at any finite d, and at d-->infinity the transition is of first order. The typical rms amplitude of a quasispecies around the master sequence is found to diverge algebraically with exponent nu(perpendicular to)=1 at the transition to the delocalized phase in the limit d-->infinity.