Asymptotic properties of the equilibrium probability of identity in a geographically structured population

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Abstract
LetI(x, u) be the probability that two genes found a vector distancexapart are the same type in an infinite-allele selectively-neutral migration model with mutation rateu.The creatures involved inhabit an infinite of colonies, are diploid and are held atNper colony. Setin one dimension andin higher dimensions, where σ2is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Thenin one dimension,in two dimensions, andin three dimensions uniformly forHereC0is a constant depending on the migration law,K0(y) is the Bessel function of the second kind of order zero, andare the eigenvalues of σ2. For symmetric nearest-neighbor migrations,in one dimension andlogmiin two. Foris known in one dimension andC0does not appear. In two dimensions,These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

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