Renormalisation of hierarchically interacting Cannings processes

Abstract

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. After that we introduce a system of hierarchically interacting $C^\Lambda$-processes, where the interaction comes from migration and reshuffling-resampling based on measures $(\Lambda_k)_{k}$ both acting in $k$-blocks of the hierarchical group. We refer to this system as the $C_N^{c,\Lambda}$-process. The dual process of the $C_N^{c,\Lambda}$-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 $N\to\infty$. Our main result is that, in the limit as $N\to\infty$, on each scale $k\in\mathbb{N}_0$ the $k$-block averages of the $C_N^{c,\Lambda}$-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 $0\leq l<k$. Thus, it is through the volatility that the renormalisation manifests itself. We discuss the implications of the scaling of $d_k$ for the behaviour on large space-time scales of the $C_N^{c,\Lambda}$-process. We compare the outcome with what is known from the renormalisation analysis of hierarchically interacting Fleming-Viot diffusions, pointing out several new features. We obtain a new classification for when the process exhibits clustering, respectively, exhibits local coexistence. Finally, we show that for finite $N$ the same dichotomy between clustering and local coexistence holds as for $N\to\infty$.