We study quasispecies and closely related evolutionary dynamics like the replicator-mutator equation in high dimensions. In particular, we show that under certain conditions, the average fitness of almost all quasispecies of a given dimension becomes independent of mutational probabilities in high dimensional sequence spaces. This result is a consequence of concentration of measure on a high dimensional hypersphere and its extension to Lipschitz functions known as the Levy's Lemma. Our results naturally extend to other functional capabilities that can be described as Lipschitz functions and whose input parameters are the frequencies of individual constituents of the quasispecies. In order to show this, we give a generalization of Levy's Lemma and discuss possible biological consequences of our work. © 2019 American Physical Society.