Coalescent approximation for structured populations in a stationary random environment

We establish convergence to the Kingman coalescent for the genealogy of a geographically–or otherwise–structured version of the Wright–Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. This brings a novel formula for the coalescent effective population size (EPS). We call it a quenched EPS to emphasize the key feature of our model — random environment. The quenched EPS is compared with an annealed (mean-field) EPS which describes the case of constant migration probabilities obtained by averaging the random migration probabilities over possible environments.