Renormalisation of hierarchically interacting Cannings processes

In order to analyse universal patterns in the large space-time behaviour of interacting multi-type stochastic populations on countable geographic spaces, a key approach has been to carry out a renormalisation analysis in the hierarchical mean-field limit. This has provided considerable insight into the structure of interacting systems of finite-dimensional diffusions, such as Fisher-Wright or Feller diffusions, and their infinite-dimensional analogues, such as Fleming-Viot or Dawson-Watanabe superdiffusions. The present paper brings a new class of interacting jump processes into focus. We start from a single-colony $C^\Lambda$-process, which arises as the continuum-mass limit of a $\Lambda$-Cannings individual-based population model, where $\Lambda$ is a finite non-negative measure that describes the offspring mechanism, i.e., how individuals in a single colony are replaced via resampling. The key feature of the $\Lambda$-Cannings individual-based population model is that the offspring of a single individual can be a positive fraction of the total population. After that we introduce a system of hierarchically interacting $C^\Lambda$-processes, where the interaction comes from migration and reshuffling-resampling on all hierarchical space-time scales simultaneously. More precisely, individuals live in colonies labelled by the hierarchical group $\Omega_N$ of order N, and are subject to migration based on a sequence of migration coefficients $\b{c}=(c_k)_{k \in N_0}$ and to reshuffling-resampling based on a sequence of resampling measures $\Lambda=(\Lambda_k)_{k \in N_0}$, both acting in k-blocks for all $k \in N_0$. The reshuffling is linked to the resampling: before resampling in a block takes place all individuals in that block are relocated uniformly, i.e., resampling is done in a locally ``panmictic'' manner. We refer to this system as the $C^{\frac{\b{c};\Lambda}{N}}$-process. The dual process of the $C^\Lambda$-process is the $\Lambda$-coalescent, whereas the dual process of the $C^{\frac{\b{c};\Lambda}{N}}$-process is a spatial coalescent with multi-level block coalescence. For the above system we carry out a full renormalisation analysis in the hierarchical mean-field limit $N \rightarrow \infty$. Our main result is that, in the limit as $N \rightarrow \infty$, on each hierarchical scale $k \in N_0$ the k-block averages of the $C^{\frac{\b{c};\Lambda}{N}}$-process converge to a random process that is a superposition of a $C^{\Lambda_k}$-process and a Fleming-Viot process, the latter with a volatility $d_k$ and with a drift of strength $c_k$ towards the limiting $(k+1)$-block average. It turns out that $d_k$ is a function of $c_l$ and $\Lambda_l$ for all $