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The hierarchical Cannings process in random environment

Greven A, den Hollander F, Klimovsky A (2017)

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Publication Type: Journal article, Online publication

Publication year: 2017

Journal

URI: https://arxiv.org/abs/1703.03061

Abstract

In an earlier paper, we introduced and studied a system of hierarchically interacting
measure-valued random processes that arises as the continuum limit of a large population
of individuals carrying different types. Individuals live in colonies labelled by the hierar-
chical group of order N , and are subject to migration and resampling on all hierarchical
scales simultaneously. The resampling mechanism is such that 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. Be-
fore resampling in a block takes place, all individuals in that block are relocated uniformly,
which we call reshuffling.
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, i.e.,
we work in 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 → ∞, 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. In some universality classes the growth of
the clusters depends on the realisation of the random environment.

MSC 2010: Primary 60J25, 60K35; Secondary 60G57, 60J60, 60J75, 82C28, 92D25.
Acknowledgements: AG was supported by the Deutsche Forschungsgemeinschaft (grant
DFG-GR 876/15-2), FdH was supported by the European Research Council (Advanced
Grant VARIS-267356) and by the Netherlands Organization for Scientific Research (Grav-
itation Grant NETWORKS-024.002.003), AK was supported by the Netherlands Orga-
nization for Scientific Research (grant 613.000.913). The authors are grateful to Evgeny
Verbitskiy for help with the renormalization analysis.

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How to cite

APA:

Greven, A., den Hollander, F., & Klimovsky, A. (2017). The hierarchical Cannings process in random environment. ALEA : Latin American Journal of Probability and Mathematical Statistics.

MLA:

Greven, Andreas, Frank den Hollander, and Anton Klimovsky. "The hierarchical Cannings process in random environment." ALEA : Latin American Journal of Probability and Mathematical Statistics (2017).

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