A stochastic model for genetic linkage equilibrium.

Linkage equilibrium is an independence condition among the alleles at a set of gene loci. Equilibrium or disequilibrium only makes sense relative to some reference population of a species. If the loci all occur on the same chromosome, then linkage equilibrium holds provided a random representative of that chromosome from the reference population displays independent alleles at the various loci of the set. Classical deterministic population genetics theory shows that linkage equilibrium is approached asymptotically after many generations of random mating in a reference population of infinite size. The current paper considers a Markov chain model for the establishment of linkage equilibrium in a population of finite size. The states of this Markov chain correspond to counts of chromosomes of various types. Because the chain is reversible, the equilibrium distribution can be explicitly computed. Partial characterization of the geometric rate of convergence of the chain to equilibrium is possible using a strong stationary stopping time.