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A branching process in varying environment (BPVE) is a Galton-Watson tree whose offspring distribution can change at each generation. The evolution of the size of successive generations when the process does not die out early has drawn a lot of attention in recent years, both from the discrete and continuum points of view (the scaling limit being a modified Continuous State Branching Process).
We focus on the limiting genealogical structure, in the critical case (all distributions have offspring mean 1). We show that under mild second moment assumptions on the sequence of offspring distributions, a BPVE conditioned to be large converges to the Brownian Continuum Random Tree, as in the standard Galton-Watson setting. The varying environment adds asymmetry and dependencies in many places. This requires numerous changes to the usual arguments, in particular for the height process, for which we propose a simple and (to our knowledge) new interpretation.
This is joint work with Daniel Kious and Cécile Mailler.
