  We study quasi-species and closely related evolutionary dynamics like the
replicator-mutator equation in high dimensions. In particular, we show that
under certain conditions 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 known %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. We discuss the consequences of our study
for the replicator-mutator equation.
