  We show that, under certain assumptions, the fitness of almost all
quasi-species becomes independent of mutational probabilities and the initial
frequency distributions of the sequences in high dimensional sequence spaces.
This result is the consequence of the concentration of measure on a high
dimensional hypersphere and its extension to Lipschitz functions knows as the
Levy's Lemma. Therefore, evolutionary dynamics almost always yields the same
value for fitness of the quasi-species, independent of the mutational process
and initial conditions, and is quite robust to mutational changes and
fluctuations in initial conditions. Our results naturally extend to any
Lipschitz function whose input parameters are the frequencies of individual
constituents of the quasi-species. This suggests that the functional
capabilities of high dimensional quasi-species are robust to fluctuations in
the mutational probabilities and initial conditions.
