Matthias Birkner (Johannes Gutenberg University Mainz, Germany)

Ancestral lineages under local regulation

The spatial embeddings of genealogies in models with fluctuating population sizes and local regulation are relatively complicated random walks in a space-time dependent random environment. They seem presently not well understood. We use the supercritical discrete-time contact process on Z^d as the simplest non-trivial example of a locally regulated population model and study the dynamics of an ancestral lineage sampled at stationarity, viz. a directed random walk on a supercritical directed percolation cluster. We prove a law of large numbers and an annealed central limit theorem (CLT) for such a walk via a regenerative approach. Analysis of the joint dynamics of two walks in the same medium allows to obtain also a quenched CLT, at least in the case d≥3.  Furthermore, we discuss approaches to extend these results to more general models that allow multiple occupancy of sites and implications for the spatial distribution of neutral types in equilibrium.

Based on joint work in progress with Jiří Černý, Andrej Depperschmidt and Nina Gantert.