Branching processes are a natural class of tree models where particles, that can have a heritable type, reproduce independently in a way that depends on their type. In the critical regime, these trees display an interesting limiting behavior. After an appropriate rescaling, they converge to the celebrated Brownian Continuum Random Tree (CRT). Convergence to the CRT is well understood for Galton-Watson processes, but deriving it for more complex models remains a significantly harder task. In this talk, I will discuss a new general approach to prove this type of result. It relies on two main ingredients: the Gromov-weak topology and a many-to-few formula. I will give a gentle introduction to random metric spaces and to the Gromov-weak topology, and present a version of the many-to-few formula. It uses an ingenious random change of measure to compute the so-called moments of the tree. Finally, I will illustrate these concepts with an application to a class of branching processes with a mixing behavior.
Félix is a Glasstone Fellow at the Department of Statistics of the University of Oxford. Before that, he did a one-year postdoc at the UQAM in Montréal, and completed his PhD in Paris, jointly at the LPSM and Collège de France. He has a mixed background in evolutionary biology and probability theory, and his research lies at the interface between these two fields.
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