In 2002, Duquesne and Le Gall established an invariance principle for discrete Galton-Watson forests, characterizing their scaling limits as a class of continuum random trees called Lévy forests, in an analogous manner to how continuous-state branching processes are obtained as scaling limits of discrete Galton-Watson processes. Their invariance principle, however, relied on two assumptions: i) that the Galton-Watson forests are subcritical, and ii) that Grey’s condition is satisfied in the limit. The invariance principle has since been extended in work by Duquesne and Winkel (2019, 2025+) leaving open only the case where both assumptions i) and ii) fail. In this case limiting trees may be both unbounded and not locally compact, and as such the classical Gromov-type topologies used for studying convergence of R-trees are not suitable. We develop instead a weaker notion of convergence by extending the technique of mass erasure to R-trees equipped with a suitable class of boundedly finite measures.

The talk is based on ongoing work, and its aim will be to introduce the invariance principles, the notion of mass erasure and some of the main results of the project. No prior knowledge is assumed, other than standard probability- and measure theory.