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

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  • Ojeda, Gabriel Berzunza
  • Holmgren, Cecilia
  • Thévenin, Paul

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.

Suggested Citation

  • Ojeda, Gabriel Berzunza & Holmgren, Cecilia & Thévenin, Paul, 2026. "Convergence of trees with a given degree sequence and of their associated laminations," Stochastic Processes and their Applications, Elsevier, vol. 196(C).
    • Handle: RePEc:eee:spapps:v:196:y:2026:i:c:s0304414925002601
    DOI: 10.1016/j.spa.2025.104816
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