= Solution
Five consequences of the <ancestral recombination graph> are:
* \b[<Linkage disequilibrium> and its distance dependence.] Nearby sites often inherit the same local genealogy, so <mutations> on their shared ancestral branches produce correlated <alleles>. The chance of an ancestral breakpoint between sites increases with their genetic distance, reducing shared genealogy and, on average, <linkage disequilibrium>. This gives the genealogical explanation of the two-locus decay calculation.
* \b[<Haplotype blocks> and <recombination hotspots>.] With a varying recombination map, the <ARG> has relatively few breakpoints within low-recombination regions and many near <recombination hotspots>. Long intervals can consequently retain a small number of common ancestral <haplotypes>, while genealogical relationships change more frequently across the intervening regions. This explains why LD blocks are heterogeneous rather than equally spaced units.
* \b[Mosaic inheritance and shared ancestral tracts.] Backward splits assign different pieces of one contemporary <chromosome> to different ancestral <chromosomes>. Two sampled <chromosomes> can therefore share a recent ancestor over one interval but not its neighbors. For a specified total of $m$ independent ancestral <meioses>, a tract of genetic length $d$ has approximately $md$ expected crossovers under a Poisson crossover model, so its <probability> of remaining intact is $e^{-md}$. This relates long <IBD segments> to recent common ancestry, while distinguishing tract-boundary sampling from length-biased sampling at a random genomic point.
* \b[Incompatible site patterns and four observed gametes.] Under an <infinite sites mutation model> on a single rooted tree, the descendant sets of two <mutations> are nested or disjoint. They cannot cross, so all four <haplotypes> $00,01,10,11$ cannot occur. Recombination permits different local trees: one site's derived clade can be $\{10,11\}$ and the other's $\{01,11\}$, producing all four without repeated <mutation>. The <four-gamete test> therefore detects a failure of the unrecombined, single-mutation-per-site model. Recurrent <mutation> would be an alternative explanation if the infinite-sites assumption were dropped.
* \b[Variation in local diversity and <allele> frequencies.] Coalescent mergers describe <genetic drift>; their random times produce different branch lengths and descendant-group sizes. Recombination makes these genealogies vary along the <chromosome>. Conditional on a local genealogy, its <segregating site> count is Poisson with <mean> $\theta L/2$, so a longer tree tends to contain more variants. A <mutation> on a branch subtending $j$ of the $n$ samples appears at sample frequency $j/n$. Together, random branch lengths, descendant counts and <mutations> explain heterogeneous local diversity and the distribution of <allele> frequencies without requiring selection.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-45-recombination-genealogies.png]
{title=Different local genealogies can generate all four haplotypes with one mutation per site}
\b[Shared ancestry explains <correlation>; recombination makes ancestry a mosaic; <mutation> on the resulting branches converts genealogy into observed variation.] These explanations use the neutral ARG, with a variable recombination map where stated. Selection, population subdivision or changing population size can also be studied genealogically, but require extending the event model rather than being consequences of the constant-size panmictic neutral process alone.
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