随机环境中的递阶Cannings过程
The hierarchical Cannings process in random environment
摘要：
在以前的论文中，我们介绍和研究了一个系统的层次相互作用的测度值随机过程，它描述了一个大的人口携带类型和生活在殖民地标记的层次组的顺序$N$的个人。这些个体同时在所有等级尺度上迁移、重新定居。在重新分配时，一个菌落块中的随机阳性部分群体继承了该块中随机单个个体的类型，这就是为什么我们将我们的系统称为分层Cannings过程。在本文中，我们研究了一个版本的分层Cannings过程在随机环境中，即，控制类型的变化在不同的块中的个人的reservation措施是随机选择与一个给定的平均值，并保持固定的时间（=淬火设置）。我们给出了接近多类型均衡（=共存）而不是单类型均衡（=聚集）的充要条件。此外，在层次平均场极限$N \to \infty$，与重整化分析的帮助下，我们得到一个完整的图片的空间-时间标度行为的块平均在所有层次尺度上同时。我们发现，$k$块平均分布的叠加与确定性的波动率常数$d_k$和一个Cannings过程与随机跳跃率的弗莱明-维奥特扩散，都依赖于$k$。在随机环境中，$d_k$证明是小于在均匀环境中相同的平均值。我们研究了$d_k$如何与$k$一起扩展。这导致五个普遍性类的集群形成的单型制度。我们发现，如果集群发生，那么随机环境减缓了集群的增长，即，增加了类型的多样性。
英文摘要：
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.
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