From Cannings model to Brownian motion conditioned on local time profile

Abstract

We study the scaling limits of genealogical trees arising from Cannings models. Under suitable moment conditions, we show that the rescaled contour and height functions converge to a time change of Brownian motion conditioned on a given local time profile. This conditioned Brownian motion is a self-interacting diffusion constructed independently by Warren--Yor (1998) and Aldous (1998). A key ingredient in our proof is a sequential version of the coming-down-from-infinity property.