= Linkage-disequilibrium decay under recombination
{title2=$D_{t+1}=(1-r)D_t$}
<Genetic recombination> erodes <linkage disequilibrium> by reshuffling <alleles> between <haplotypes>, while preserving marginal <allele frequencies> in the idealized model. Let $f_{AB,t}$ denote the <haplotype frequency> in the <gamete> pool of generation $t$, let $p,q$ denote its marginal <allele frequencies>, and let $r\in[0,1/2]$ be the constant <recombination fraction> between two autosomal <genetic loci>. Assume <random mating> within one population, nonoverlapping generations, fair <Mendelian segregation>, equal reciprocal <gamete> production, the same $r$ in both sexes, and a deterministic population large enough to neglect <genetic drift>. Exclude <natural selection>, <mutation> and <gene flow>. Then
$$
f_{AB,t+1}=(1-r)f_{AB,t}+rpq=f_{AB,t}-rD_t,
\qquad D_{t+1}=(1-r)D_t,
\qquad D_t=(1-r)^tD_0.
$$
Here $p$ and $q$ remain constant: recombining changes which <alleles> occur together, not their total numbers. The other <haplotype frequencies> change by $+rD_t$ for $Ab,aB$ and $-rD_t$ for $ab$. Consequently the <joint probability distribution> stays within its allowed bounds and approaches the product of its <marginal distributions> when $r>0$.
For the original positive example $p=0.60,q=0.40,D_0=0.12$, a <recombination fraction> $r=0.10$ gives $D_1=0.108$ and $D_{10}\approx0.04184$. The negative example $D_0=-0.12$ has exactly the opposite sequence: <genetic recombination> reduces the magnitude without reversing the sign. With $r=0$, the initial <linkage disequilibrium> persists in this model; even <independent assortment> at $r=1/2$ only halves $D$ each generation. A population association does not vanish after one generation merely because the <genetic loci> are unlinked.
\Image[/linkage-disequilibrium-ideal-recombination-decay.png]
{title=Idealized <linkage-disequilibrium decay under recombination> at four <recombination fractions>, with faster decay of squared <allelic correlation> when marginal <allele frequencies> remain fixed}
The <geometric sequence> gives a generation scale: for $r>0$, the continuous interpolation reaches $|D_0|/2$ at $t_{1/2}=\log(1/2)/\log(1-r)$. Actual generations are integers; the first at or below half is $\lceil t_{1/2}\rceil$. For small $r$, $t_{1/2}\approx(\log 2)/r$. Because the denominator of <allelic correlation> stays fixed, $r_{\mathrm{LD},t}^2=(1-r)^{2t}r_{\mathrm{LD},0}^2$; its decay is faster than the magnitude of $D_t$.
A short derivation uses <random mating> to assign frequency $2f_{AB}f_{ab}$ to the double <heterozygote> $AB/ab$ and $2f_{Ab}f_{aB}$ to $Ab/aB$. Relative to $r=0$, each loses or gains an $AB$ <gamete> fraction $r/2$, respectively. Their net contribution is $r(f_{Ab}f_{aB}-f_{AB}f_{ab})=-rD$. The other <genotypes> produce no net change in $f_{AB}$, proving the recurrence and, by iteration, its solution.
This temporal model does not identify elapsed generations from physical separation alone. The <recombination fraction> depends on <genetic recombination>, including <recombination hotspots>, while <population stratification>, <natural selection>, <gene flow> or <genetic drift> can create or maintain <linkage disequilibrium>. Repeated sampling across time would also require accounting for <sample size> and <haplotype estimation>. A cross-sectional plot of association against distance between <genetic loci> is therefore a different observation from the idealized trajectory plotted here.
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