  The parallel mutation-selection evolutionary dynamics, in which mutation and
replication are independent events, is solved exactly in the case that the
Malthusian fitnesses associated to the genomes are described by the Random
Energy Model (REM) and by a ferromagnetic version of the REM. The solution
method uses the mapping of the evolutionary dynamics into a quantum Ising chain
in a transverse field and the Suzuki-Trotter formalism to calculate the
transition probabilities between configurations at different times. We find
that in the case of the REM landscape the dynamics can exhibit three distinct
regimes: pure diffusion or stasis for short times, depending on the fitness of
the initial configuration, and a spin-glass regime for large times. The dynamic
transition between these dynamical regimes is marked by discontinuities in the
mean-fitness as well as in the overlap with the initial reference sequence. The
relaxation to equilibrium is described by an inverse time decay. In the
ferromagnetic REM, we find in addition to these three regimes, a ferromagnetic
regime where the overlap and the mean-fitness are frozen. In this case, the
system relaxes to equilibrium in a finite time. The relevance of our results to
information processing aspects of evolution is discussed.
