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We show joint convergence of the Łukasiewicz path and height process for slightly supercritical Galton–Watson forests. This shows that the height processes for supercritical continuous-state branching processes as constructed by Lambert (2002) are the limit under rescaling of their discrete counterparts. Unlike for (sub-)critical Galton–Watson forests, the height process does not encode the entire metric structure of a supercritical Galton–Watson forest. We demonstrate that this result is nonetheless useful, by applying it to the configuration model with an independent and identically distributed power-law degree sequence in the critical window, of which we obtain the metric space scaling limit in the product Gromov–Hausdorff–Prokhorov topology, which is of independent interest.

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Convergence of the height process of supercritical Galton–Watson forests with an application to the configuration model in the critical window

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Convergence of the height process of supercritical Galton–Watson forests with an application to the configuration model in the critical window