The infinite rate symbiotic branching model: from discrete to continuous space

The symbiotic branching model describes a spatial population consisting of two types that are allowed to migrate in space and branch locally only if both types are present. We continue our investigation of the large scale behaviour of the system started in Blath, Hammer and Ortgiese (2016), where we showed that the continuum system converges after diffusive rescaling. Inspired by a scaling property of the continuum model, a series of earlier works initiated by Klenke and Mytnik (2010, 2012) studied the model on a discrete space, but with infinite branching rate. In this paper, we bridge the gap between the two models by showing that by diffusively rescaling this discrete space infinite rate model, we obtain the continuum model from Blath, Hammer and Ortgiese (2016). As an application of this convergence result, we show that if we start the infinite rate system from complementary Heaviside initial conditions, the initial ordering of types is preserved in the limit and that the interface between the types consists of a single point.Comment: 36 pages, 1 figur

The largest strongly connected component in Wakeley et al's cyclical pedigree model

We establish a link between Wakeley et al's (2012) cyclical pedigree model from population genetics and a randomized directed configuration model (DCM) considered by Cooper and Frieze (2004). We then exploit this link in combination with asymptotic results for the in-degree distribution of the corresponding DCM to compute the asymptotic size of the largest strongly connected component $S^N$ (where $N$ is the population size) of the DCM resp. the pedigree. The size of the giant component can be characterized explicitly (amounting to approximately $80 \%$ of the total populations size) and thus contributes to a reduced `pedigree effective population size'. In addition, the second largest strongly connected component is only of size $O(\log N)$. Moreover, we describe the size and structure of the `domain of attraction' of $S^N$. In particular, we show that with high probability for any individual the shortest ancestral line reaches $S^N$ after $O(\log \log N)$ generations, while almost all other ancestral lines take at most $O(\log N)$ generations.Comment: 21 pages, 2 figure