From Cannings model to Brownian motion conditioned on local time profile

TL;DR

This paper investigates the scaling limits of genealogical trees from Cannings models, showing convergence to a conditioned Brownian motion with a specific local time profile, revealing new connections between population models and self-interacting diffusions.

Contribution

It establishes the convergence of genealogical tree functions to a conditioned Brownian motion, linking population genetics models with advanced stochastic processes.

Findings

Rescaled contour and height functions converge to conditioned Brownian motion.

The conditioned process is a self-interacting diffusion independently constructed by Warren--Yor and Aldous.

A sequential coming-down-from-infinity property is key to the proof.

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.