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Phase Transition for Tree-Rooted Maps

Authors Marie Albenque, Éric Fusy, Zéphyr Salvy

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Author Details

Marie Albenque
  • IRIF, Université Paris Cité, France
Éric Fusy
  • Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France
Zéphyr Salvy
  • Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France

Funding

ANR CartesEtPlus "ANR-23-CE48-0018".
  • Albenque, Marie: ANR "IsOMa" under the agreeement ANR-21-CE48-0007.

Acknowledgements

We would like to thank the anonymous referees for their careful reading and their insightful comments.

Cite AsGet BibTex

Marie Albenque, Éric Fusy, and Zéphyr Salvy. Phase Transition for Tree-Rooted Maps. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.6

Abstract

We introduce a model of tree-rooted planar maps weighted by their number of 2-connected blocks. We study its enumerative properties and prove that it undergoes a phase transition. We give the distribution of the size of the largest 2-connected blocks in the three regimes (subcritical, critical and supercritical) and further establish that the scaling limit is the Brownian Continuum Random Tree in the critical and supercritical regimes, with respective rescalings √{n/log(n)} and √n.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Stochastic processes
  • Mathematics of computing → Probability and statistics
Keywords
  • Asymptotic Enumeration
  • Planar maps
  • Random trees
  • Phase transition

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