Scaling limits of random Pólya trees

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Abstract

Pólya trees are rooted trees considered up to symmetry. We establish the convergence of large uniform random Pólya trees with arbitrary degree restrictions to Aldous’ Continuum Random Tree with respect to the Gromov–Hausdorff metric. Our proof is short and elementary, and it is based on a novel decomposition: it shows that the global shape of a random Pólya tree is essentially dictated by a large Galton–Watson tree that it contains. We also derive sub-Gaussian tail bounds for both the height and the width, which are optimal up to constant factors in the exponent.

Keywords

Random trees Scaling limits Pólya trees 

Mathematics Subject Classification

60F17 60C05 05C05 

Notes

Acknowledgements

We thank two anonymous referees for thoroughly reading the manuscript and giving helpful suggestions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institute of MathematicsLudwig-Maximilians-Universität MunichMunichGermany
  2. 2.Unité de Mathématiques Pures et AppliquéesÉcole Normale Supérieure de LyonLyonFrance

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