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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

Part of: Markov processes Combinatorial probability

Published online by Cambridge University Press:  02 April 2024

Serte Donderwinkel Open the ORCID record for Serte Donderwinkel [Opens in a new window]
Serte Donderwinkel*
Affiliation:
McGill University
*
*Postal address: Burnside Hall, 805 Sherbrooke Street West, Montréal, Quebec H3A 0B9, Canada. Email address: serte.donderwinkel@mcgill.ca
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Abstract

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.

Keywords

Random treesbranching processesGalton–Watson treesBienaymé treesmetric space convergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60C05: Combinatorial probability
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 42
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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