Before revision:  Moreover, after normalization, the limiting fluctuations are shown to be Gaussian with a covariance function in the replicator equation's solution through the Wright--Fisher noise coefficient. In particular, to the degree of using constants that may depend implicitly on space, we prove a conjecture from the biological literature that the expected density processes approximate the replicator equation on many non-regular graphs.
After revision:  Moreover, after normalization, the limiting fluctuations are shown to be Gaussian with a covariance function in the limiting densities. As an application, we resolve in the positive a conjecture from the biological literature that the expected density processes approximate the replicator equation on many non-regular graphs.