The Annals of Probability

Results for the Stepping Stone Model for Migration in Population Genetics

Stanley Sawyer

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The stepping stone model describes a situation in which beasts alternately migrate among an infinite array of colonies, undergo random mating within each colony, and are subject to selectively neutral mutation at the rate $u$. Assume the beasts follow a random walk $\{X_n\}$. If $u = 0$, we show that two randomly chosen beasts in the $n$th generation in any bounded set are genetically identical at a given locus with probability converging to one iff the symmetrization of $\{X_n\}$ is recurrent. In general, if either $u = 0$ or $u$ is of order $1/n$, this probability converges to its limit at the rate $C/n^{\frac{1}{2}}$ for finite variance walks in one dimension and $C/(\log n)^a$ in two, with other rates for other classes of $\{X_n\}$. More complicated rates ensure for $u \neq O(1/n)$.

Article information

Ann. Probab. Volume 4, Number 5 (1976), 699-728.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 92A10
Secondary: 92A15 60J15 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 60K99: None of the above, but in this section

Stepping stone model random walks genetics population genetics diploid migration mutation random mating rate of convergence


Sawyer, Stanley. Results for the Stepping Stone Model for Migration in Population Genetics. Ann. Probab. 4 (1976), no. 5, 699--728. doi:10.1214/aop/1176995980.

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