The distribution of the time to the most recent common genetic ancestor of a pair of genomes

For every gene in a diploid genome, the maternal and paternal copies share a common ancestor at some point in the past. Because of recombination, each gene potentially has its own ancestral history and time to the most recent common ancestor (TMRCA). Basic evolutionary theory predicts that the marginal distribution of the TMRCA for each gene is exponential with a mean proportional to the population size, but the joint distribution across genes is unknown. We use asymptotic analysis and simulations to find simple expressions for the distribution of the minimum TMRCA across the genome. We find that the minimum TMRCA is tightly peaked around a value that is proportional to the square root of the population size. For humans, this means that your most closely related pair of genes have a TMRCA that is more than two orders of magnitude smaller than a typical pair.