In the present article, we investigate the effects of dormancy on an abstract population genetic level. We first provide a short review of seed bank models in population genetics, and the role of dormancy for the interplay of evolutionary forces in g
eneral, before we discuss two recent paradigmatic models, referring to spontaneous resp. simultaneous switching of individuals between the active and the dormant state. We show that both mechanisms give rise to non-trivial mathematical objects, namely the (continuous) seed bank diffusion and the seed bank diffusion with jumps, as well as their dual processes, the seed bank coalescent and the seed bank coalescent with simultaneous switching.
We introduce a new Wright-Fisher type model for seed banks incorporating simultaneous switching, which is motivated by recent work on microbial dormancy. We show that the simultaneous switching mechanism leads to a new jump-diffusion limit for the sc
aled frequency processes, extending the classical Wright-Fisher and seed bank diffusion limits. We further establish a new dual coalescent structure with multiple activation and deactivation events of lineages. While this seems reminiscent of multiple merger events in general exchangeable coalescents, it actually leads to an entirely new class of coalescent processes with unique qualitative and quantitative behaviour. To illustrate this, we provide a novel kind of condition for coming down from infinity for these coalescents using recent results of Griffiths.
We consider the real-valued centered Gaussian field on the four-dimensional integer lattice, whose covariance matrix is given by the Greens function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. $d=4$ is the
critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field.
Consider the centered Gaussian field on the lattice $mathbb{Z}^d,$ $d$ large enough, with covariances given by the inverse of $sum_{j=k}^K q_j(-Delta)^j,$ where $Delta$ is the discrete Laplacian and $q_j in mathbb{R},kleq jleq K,$ the $q_j$ satisfyin
g certain additional conditions. We extend a previously known result to show that the probability that all spins are nonnegative on a box of side-length $N$ has an exponential decay at rate of order $N^{d-2k}log{N}.$ The constant is given in terms of a higher-order capacity of the unit cube, analogous to the known case of the lattice free field. This result then allows us to show that, if we condition the field to stay positive in the $N-$box, the local sample mean of the field is pushed to a height of order $sqrt{log N}.$
