Steady state of homozygosity and GstGst for the island model

We examine homozygosity and GstGst for a subdivided population governed by the finite island model. Assuming an infinite allele model and strong mutation we show that the steady state distributions of GstGst and homozygosity have asymptotic expansions in the mutation rate. We use this observation to derive asymptotic expansions for various moments of homozygosity and to derive rigorous formulas for the mean and variance of GstGst. We show that Gst≈1/(1+2Nm)Gst≈1/(1+2Nm), similarly to the well known formula of Wright for the infinite island model, and that the variance of GstGst goes to zero as mutation increases.