Convergence of trees with a given degree sequence and of their associated laminations

Gabriel Berzunza Ojeda, Cecilia Holmgren and Paul Thévenin

Stochastic Processes and their Applications, 2026, vol. 196, issue C

Abstract: In this paper, we study uniform rooted plane trees with given degree sequence. We show that, under some natural hypotheses on the degree sequences, these trees converge towards the so-called Inhomogeneous Continuum Random Tree after renormalization. Our proof relies on the convergence of a modification of the well-known Łukasiewicz path. We also give a unified treatment of the limit, as the number of vertices tends to infinity, of the fragmentation process derived by cutting down the edges of a tree with a given degree sequence, including its geometric representation by a lamination-valued process. The latter is a collection of nested laminations, which are compact subsets of the unit disk made of non-crossing chords. In particular, we prove an equivalence between planar Gromov-weak convergence of discrete trees and the convergence of their associated lamination-valued processes.

Keywords: Bridge with exchangeable increments; Continuum random tree; Fragmentation processes; Inhomogeneous CRT; Lamination of the disk; Scaling limits (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1016/j.spa.2025.104816

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