The hierarchical Cannings process in random environment

Abstract: In an earlier paper, we introduced and studied a system of hierarchically interacting measure-valued random processes which describes a large population of individuals carrying types and living in colonies labelled by the hierarchical group of order $N$. The individuals are subject to migration, resampling on all hierarchical scales simultaneously. Upon resampling, a random positive fraction of the population in a block of colonies inherits the type of a random single individual in that block, which is why we refer to our system as the hierarchical Cannings process. In the present paper, we study a version of the hierarchical Cannings process in random environment, namely, the resampling measures controlling the change of type of individuals in different blocks are chosen randomly with a given mean and are kept fixed in time (= the quenched setting). We give a necessary and sufficient condition under which a multi-type equilibrium is approached (= coexistence) as opposed to a mono-type equilibrium (= clustering). Moreover, in the hierarchical mean-field limit $N \to \infty$, with the help of a renormalization analysis, we obtain a full picture of the space-time scaling behaviour of block averages on all hierarchical scales simultaneously. We show that the $k$-block averages are distributed as the superposition of a Fleming-Viot diffusion with a deterministic volatility constant $d_k$ and a Cannings process with a random jump rate, both depending on $k$. In the random environment, $d_k$ turns out to be smaller than in the homogeneous environment of the same mean. We investigate how $d_k$ scales with $k$. This leads to five universality classes of cluster formation in the mono-type regime. We find that if clustering occurs, then the random environment slows down the growth of the clusters, i.e., enhances the diversity of types.