Pairwise Shared Ancestry in Random-Mating Constant-Size Populations

In a panmictic population of constant size N, random pairs of individuals will have a most recent shared ancestor who lived slightly more than 0.5 log 2 N generations previously, on average. The probability that a random pair of individuals will share at least one ancestor who lived 0.5 log 2 N generations ago, or more recently, is about 50%. Those individuals, if they do share an ancestor from that generation, would be cousins of degree (0.5 log 2 N) - 1. Shared ancestry from progressively earlier generations increases rapidly until there is universal pairwise shared ancestry. At that point, every individual has one or more ancestors in common with every other individual in the population, although different pairs may share different ancestors. Those ancestors lived approximately 0.7 log 2 N generations in the past, or more recently. Qualitatively, the ancestries of random pairs have about 50% similarity for ancestors who lived about 0.9 log 2 N generations before the present. That is, about half of the ancestors from that generation belonging to one member of the pair are present also in the genealogy of the other member. Qualitative pairwise similarity increases to more than 99% for ancestors who lived about 1.4 log 2 N generations in the past. Similar results apply to a metric of quantitative pairwise genealogical overlap.