On Stabilizing the Rate of Isonymy Divergence

A theoretical analysis of the surname state of a population (the concentration vector of namesakes in the male component of the population) and its dynamics as a result of random surname drift is carried out. An approximation of this process by the Wright–Fisher model of a population with non-overlapping generations, not subjected to selective pressure, i.e., a sequence of nested random samplings with replacement from a set of parental surnames, is used. The sample size is equal to N/2 according to the size of the male component in a population of size N. In the same population, the processes of random drift of both surnames and genes occur simultaneously. Their principal difference is that the sample size of surnames is four times smaller than that of alleles at autosomal locus. The analysis of random drift is simplified by the transition from coordinates-concentrations to their square roots. Under generational change, the state receives a sample deviation measured by angular distance, and its mean square gives the divergence rate stabilizing in the new coordinates. An adaptation (relative to the analysis of the surname drift) of known in population genetics results on the nature of divergence at the stage of relatively small number of generations compared to the size of the population is given. The surname divergence occurs 4 times faster than divergence of allele concentrations.