Volume 9 (2026)
Blanc-Renaudie, Arthur;  Kammerer, Emmanuel
Scaling limit of trees with vertices of fixed degrees and heights
Annales Henri Lebesgue, Volume 9 (2026), pp. 423-469

Metadata

DOI10.5802/ahl.270
Keywords random trees ,  scaling limits ,  coalescent processes ,  branching processes ,  varying environment

Abstract

We consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalized converge. To this end, we study the paths from random vertices to the root using coalescent processes. As an application, we obtain scaling limits of Bienaymé–Galton–Watson trees in varying environment.


References

[ADH13] Abraham, Romain; Delmas, Jean-François; Hoscheit, Patrick A note on the Gromov–Hausdorff–Prokhorov distance between (locally) compact metric measure spaces, Electron. J. Probab., Volume 18 (2013), 14, 21 pages | DOI | Zbl | MR

[Ald91] Aldous, David The Continuum Random Tree. I, Ann. Probab., Volume 19 (1991) no. 1, pp. 1-28 | DOI | Zbl | MR

[Ald93] Aldous, David The Continuum Random Tree. III, Ann. Probab., Volume 21 (1993) no. 1, pp. 248-289 | DOI | Zbl | MR

[AP00] Aldous, David; Pitman, Jim Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent, Probab. Theory Relat. Fields, Volume 118 (2000) no. 4, pp. 455-482 | DOI | Zbl | MR

[AUB20] Angtuncio, Osvaldo; Uribe Bravo, Gerónimo On the profile of trees with a given degree sequence (2020) | arXiv | Zbl

[BBRKK25a] Bellin, Étienne; Blanc-Renaudie, Arthur; Kammerer, Emmanuel; Kortchemski, Igor Uniform attachment with freezing, Ann. Appl. Probab., Volume 35 (2025) no. 4, pp. 2882-2922 | DOI | Zbl | MR

[BBRKK25b] Bellin, Étienne; Blanc-Renaudie, Arthur; Kammerer, Emmanuel; Kortchemski, Igor Uniform attachment with freezing: Scaling limits, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 61 (2025) no. 4, pp. 2679-2708 | DOI | Zbl | MR

[Ber06] Bertoin, Jean Random Fragmentation and Coagulation Processes, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2006 | DOI | Zbl | MR

[BFRS22] Boenkost, Florin; Foutel-Rodier, Félix; Schertzer, Emmanuel The genealogy of nearly critical branching processes in varying environment (2022) | arXiv | Zbl

[BM14] Broutin, Nicolas; Marckert, Jean-François Asymptotics of trees with a prescribed degree sequence and applications, Random Struct. Algorithms, Volume 44 (2014) no. 3, pp. 290-316 | DOI | Zbl | MR

[BR21] Blanc-Renaudie, Arthur Limit of trees with fixed degree sequence (2021) | arXiv | Zbl

[BS15] Bansaye, Vincent; Simatos, Florian On the scaling limits of Galton–Watson processes in varying environments, Electron. J. Probab., Volume 20 (2015), 75, 36 pages | DOI | Zbl | MR

[CKKM24] Conchon-Kerjan, Guillaume; Kious, Daniel; Mailler, Cécile Scaling limit of critical random trees in random environment, Electron. J. Probab., Volume 29 (2024), 112, 53 pages | DOI | Zbl | MR

[CP00] Camarri, Michael; Pitman, Jim Limit distributions and random trees derived from the birthday problem with unequal probabilities, Electron. J. Probab., Volume 5 (2000) no. 2, 1, 19 pages | DOI | Zbl | MR

[DLG05] Duquesne, Thomas; Le Gall, Jean-François Probabilistic and fractal aspects of Lévy trees, Probab. Theory Relat. Fields, Volume 131 (2005) no. 4, pp. 553-603 | DOI | Zbl | MR

[FLL22] Fang, Rongjuan; Li, Zenghu; Liu, Jiawei A scaling limit theorem for Galton–Watson processes in varying environments, Proc. Steklov Inst. Math., Volume 316 (2022), pp. 137-159 | DOI | Zbl | MR

[HPP24] Harris, Simon C.; Palau, Sandra; Pardo, Juan Carlos The coalescent structure of Galton–Watson trees in varying environments, Ann. Appl. Probab., Volume 34 (2024) no. 6, pp. 5388-5425 | DOI | MR | Zbl

[JS03] Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes., Grundlehren der Mathematischen Wissenschaften, 288, Springer, 2003 | Zbl | DOI | MR

[Ker20] Kersting, Götz A unifying approach to branching processes in a varying environment, J. Appl. Probab., Volume 57 (2020) no. 1, pp. 196-220 | DOI | Zbl

[Kur78] Kurtz, Thomas G. Diffusion approximations for branching processes, Branching processes. Papers presented at a conference held in Quebec, August 11-20, 1976 (Joffe, Anatole; Ney, Peter, eds.) (Advances in Probability and Related Topics), Volume 5, Marcel Dekker (1978), pp. 269-292 | MR | Zbl

[Loe13] Loehr, Wolfgang Equivalence of Gromov–Prohorov- and Gromov’s □ ̲ λ -metric on the space of metric measure spaces, Electron. Commun. Probab., Volume 18 (2013), 17, 10 pages | DOI | Zbl | MR

[Mar18] Marzouk, Cyril Scaling limits of random bipartite planar maps with a prescribed degree sequence, Random Struct. Algorithms, Volume 53 (2018) no. 3, pp. 448-503 | DOI | Zbl | MR

[Pit06] Pitman, Jim Combinatorial stochastic processes: École d’eté de probabilités de Saint-Flour XXII-2002, Lecture Notes in Mathematics, 1875, Springer, 2006 | DOI | Zbl | MR