Scaling limits in interacting particle systems

This thesis treats the behaviour of certain Interacting Particle Systems (IPSs) with a large number of particles. In this thesis, we present and investigate four different IPSs: two come from the field of population genetics, the other two from the field of particle systems on lattices. The first objective of this thesis is to study the influence of fluctuating population sizes on the evolution of the gene pool. We introduce a variant of the Λ-Fleming-Viot model allowing for fluctuating population sizes and show that the process is well-defined, that the class of processes is closed under appropriate limits and that every classical Λ-Fleming-Viot process can be approximated arbitrarily well. Furthermore, we provide a lookdown construction and use it to investigate the genealogy. Secondly, we study a two-type logistic branching process with weak selection and mutation to model the regrowth of a population after a catastrophic bottleneck. We show that the genealogy before reaching carrying capacity is approximated by the genealogy of the corresponding independent branching process without mutation or selection. We further prove that the frequency process at large carrying capacity is approximated by the corresponding Gillespie-Wright-Fisher equation and provide a moment dual. The second part of the thesis is concerned with the study of certain particle systems on lattices, with particular focus on systems in contact with stochastic reservoirs. More precisely, the thesis investigates separately the influence on long-range dynamics on the one side and the effect of weak reservoirs on the other side. The first case study leverages the long-range interaction in combination with the algebraic structure of the generator of generalised exclusion processes to derive the hydrodynamic characterisation without relying on the invariant measures of the process. A particular consequence is that this new approach can handle a large class of jump kernels that are not translation invariant. The second result concerns the class of misanthrope processes: it proves that interactions with stochastic reservoirs are egligible up to the level of large deviations as soon as they occur at a slower time scale than the bulk dynamics. In the setting of one-dimensional totally asymmetric dynamics, this allows one to compare the system in contact with reservoirs with the system on Z.