The multitype allele tree for a finite-allele neutral mutation model converges in the rare-mutation limit to Bertoin's universal allele tree with deterministic type labels.
Crossing bridges between percolation models and Bienaym\'e-Galton-Watson trees
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abstract
In this survey, we explore the connections between two areas of probability: percolation theory and population genetic models. Our first goal is to highlight a construction on Galton-Watson trees, which has been described in two different ways: Bernoulli bond percolation and neutral mutations. Next, we introduce a novel connection between the Divide-and-Color percolation model and a particular multi-type Galton-Watson tree. We provide a gentle introduction to these topics while presenting an overview of the results that connect them.
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Allele trees for the mother-dependent neutral mutations model and their scaling limits in the rare mutations regime
The multitype allele tree for a finite-allele neutral mutation model converges in the rare-mutation limit to Bertoin's universal allele tree with deterministic type labels.