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From Hammersley’s lines to Hammersley’s trees

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Abstract

We construct a stationary random tree, embedded in the upper half plane, with prescribed offspring distribution and whose vertices are the atoms of a unit Poisson point process. This process which we call Hammersley’s tree process extends the usual Hammersley’s line process. Just as Hammersley’s process is related to the problem of the longest increasing subsequence, this model also has a combinatorial interpretation: it counts the number of heaps (i.e. increasing trees) required to store a random permutation. This problem was initially considered by Byers et al. (ANALCO11, workshop on analytic algorithmics and combinatorics, pp 33–44, 2011) and Istrate and Bonchis (Partition into Heapable sequences, heap tableaux and a multiset extension of Hammersley’s process. Lecture notes in computer science combinatorial pattern matching, pp 261–271, 2015) in the case of regular trees. We show, in particular, that the number of heaps grows logarithmically with the size of the permutation.

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Acknowledgements

The authors warmly thank Nathanaël Enriquez for stimulating discussions on the topic.

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Authors and Affiliations

  1. Laboratoire Modal’X, Université Paris Nanterre, Nanterre, France

    A.-L. Basdevant

  2. CMAP, Ecole Polytechnique, Palaiseau, France

    L. Gerin

  3. Laboratoire de Mathématiques, Université de Tours, Tours, France

    J.-B. Gouéré

  4. Laboratoire de Mathématiques d’Orsay, CNRS, Univ. Paris-Sud, Villejuif, France

    A. Singh

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  1. A.-L. Basdevant
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  2. L. Gerin
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  3. J.-B. Gouéré
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  4. A. Singh
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Correspondence to A. Singh.

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Basdevant, AL., Gerin, L., Gouéré, JB. et al. From Hammersley’s lines to Hammersley’s trees. Probab. Theory Relat. Fields 171, 1–51 (2018). https://doi.org/10.1007/s00440-017-0772-2

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  • DOI: https://doi.org/10.1007/s00440-017-0772-2

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