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The main object of this presentation is a branching process where each individual carries an interval that gets fragmented as time goes. It arises as the branching approximation of a Wright-Fisher model that incorporates both selection and recombination. Quite unexpectedly, this process has a rich asymptotic behaviour with two phase transitions and I will describe several of its long-term properties (survival probability, distribution of block lengths, genealogies). This behaviour is a consequence of some self-similarity of the model, and our results extend to a broad class of superprocesses sharing the same self-similarity.
I will mostly focus on the probabilistic aspects, which are work in progress with Alison Etheridge.
