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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.

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Acknowledgements

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

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Authors and Affiliations

  1. Institute of Mathematics, Ludwig-Maximilians-Universität Munich, Theresienstr. 39, 80333, Munich, Germany

    Konstantinos Panagiotou

  2. Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, 46 allée d’Italie, 69364, Lyon, France

    Benedikt Stufler

Authors
  1. Konstantinos Panagiotou
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  2. Benedikt Stufler
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Correspondence to Benedikt Stufler.

Additional information

The authors were supported by the German Research Foundation, Grant No. PA 2080/2-1 and STU 679/1-1.

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Panagiotou, K., Stufler, B. Scaling limits of random Pólya trees. Probab. Theory Relat. Fields 170, 801–820 (2018). https://doi.org/10.1007/s00440-017-0770-4

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  • DOI: https://doi.org/10.1007/s00440-017-0770-4

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