Hammond and Sheffields Power Law Pólya's Urn

Explore the second part of a 40-minute lecture on Hammond and Sheffield's power law Pólya's urn from a genealogical perspective. Delve into the case where 1/2 < α < 1, examining the genealogy of the Hammond-Sheffield urn as a random tree with vertex set Z. Discover the asymptotic depth of the most recent common ancestor (MRCA) for individuals 0 and n as n approaches infinity. Investigate attempts to analyze the depth of the MRCA of [n] using an associated hydrodynamical reaction-transport system. Learn about an analogue of the infinitely many alleles model, where individuals in Z₊ receive new types based on their parents' positions. Observe the metastable behavior of this system through simulations. This lecture, presented by Jan Lukas Igelbrink and Anton Wakolbinger at the Hausdorff Center for Mathematics, builds upon concepts introduced in the first part of the talk.