  The expected level of linkage disequilibrium (LD) in a finite ideal
population at equilibrium is of relevance for many applications in population
and quantitative genetics. Several recursion formulae have been proposed during
the last decades, whose derivations mostly contain heuristic parts and
therefore remain mathematically questionable. We propose a more justifiable
approach, including an alternative recursion formula for the expected LD. Since
the exact formula depends on the distribution of allele frequencies in a very
complicated manner, we suggest an approximate solution and analyze its validity
extensively in a simulation study. Compared to the widely used formula of Sved,
the proposed formula performs better for all parameter constellations
considered. We then analyze the expected LD at equilibrium using the theory on
discrete-time Markov chains based on the linear recursion formula, with
equilibrium being defined as the steady-state of the chain, which finally leads
to a formula for the effective population size N_e. An additional analysis
considers the effect of non-exactness of a recursion formula on the
steady-state, demonstrating that the resulting error in expected LD can be
substantial. In an application to the HapMap data of two human populations we
illustrate the dependency of the N_e-estimate on the distribution of minor
allele frequencies (MAFs), showing that the estimated N_e can vary by up to 30%
when a uniform instead of a skewed distribution of MAFs is taken as a basis to
select SNPs for the analyses. Our analyses provide new insights into the
mathematical complexity of the problem studied.
