A mathematical analysis of the stepping stone model of genetic correlation

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Abstract
In this paper we analyze the correlation coefficient between members of any two colonies out of an infinite ensemble of colonies. It is assumed that each colony has the same number of members, that migration takes place between the different colonies, and that there is a constant rate of mutation in each colony. An explicit formula is derived for the correlation function and the long distance form of this function is derived. It is shown that under rather weak restrictions on the pattern of migration the asymptotic form of the correlation function is characteristic of the dimension of the model.