Spatial Coagulation and Gelation
We consider spatial population models with local (self-)interactions. The interaction can be either induced by a common random (spatial) environment in which the individuals live or it can be modelled explicitly by a localfeed-back mechanism which regulates an individual’s offspring law depending on the configuration in a certain neighbourhood around it. These models are more realistic than classical branching random walks, where particles move and branch completely independently; they are also harder to analyse because of this lack of independence and also because they typicallyare not “monotone” systems, since adding additional individuals can be actually harmful for the present population.
We focus on (1) extinction versus survival regimes, (2) in an equilibrium, when including neutraltypes: growth of clusters versus local diversity, (3) front speed and fluctuations in expanding populations in a random environment. A common connecting tool is the study of the spatial embedding of ancestral lines, which are random walks in a dynamic random environment.
Luisa Andreis, Wolfgang Koenig, Robert Patterson:
A large-deviations principle for all the cluster sizes of a sparse Erdős-Rényi graph