Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

From Hammersley’s lines to Hammersley’s trees


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18


  1. 1.

    Aldous, D., Diaconis, P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Relat. Fields 103(2), 199–213 (1995)

  2. 2.

    Aldous, D., Diaconis, P.: Longest increasing subsequences: from patience sorting to the Baik–Deift–Johansson theorem. Bull. Am. Math. Soc. 36(4), 413–432 (1999)

  3. 3.

    Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999)

  4. 4.

    Basdevant, A.-L., Enriquez, N., Gerin, L., Gouéré, J.-B.: Discrete Hammersley’s lines with sources and sinks. ALEA 13(1), 33–52 (2016)

  5. 5.

    Byers, J., Heeringa, B., Mitzenmacher, M., Zervas, G.: Heapable sequences and subsequences. ANALCO11, Workshop on Analytic Algorithmics and Combinatorics, pp. 33–44 (2011)

  6. 6.

    Cator, E., Groeneboom, P.: Hammersley’s process with sources and sinks. Ann. Probab. 33(3), 879–903 (2005)

  7. 7.

    Groeneboom, P.: Hydrodynamical methods for analyzing longest increasing subsequences. J. Comput. Appl. Math. 142, 83–105 (2002)

  8. 8.

    Hammersley, J.M.: A few seedlings of research. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 345–394 (1972)

  9. 9.

    Istrate, G., Bonchis, C.: 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)

  10. 10.

    Kingman, J.F.C.: Subadditive ergodic theory. Ann. Probab. 1(6), 883–899 (1973)

  11. 11.

    Logan, B.F., Shepp, L.A.: A variational problem for random Young tableaux. Adv. Math. 26(2), 206–222 (1977)

  12. 12.

    Romik, D.: The Surprising Mathematics of Longest Increasing Subsequences. Cambridge University Press, Cambridge (2015)

  13. 13.

    Seppäläinen, T.: Increasing sequences of independent points on the planar lattice. Ann. Appl. Probab. 7(4), 886–898 (1997)

  14. 14.

    Seppäläinen, T.: Exact limiting shape for a simplified model of first-passage percolation on the plane. Ann. Probab. 26(3), 1232–1250 (1998)

  15. 15.

    Veršik, A.M., Kerov, S.V.: Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Dokl. Akad. Nauk SSSR 233(6), 1024–1027 (1977)

Download references


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

Author information

Correspondence to A. Singh.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Basdevant, A., Gerin, L., Gouéré, J. 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

Download citation


  • Hammersley’s process
  • Heap sorting
  • Patience sorting
  • Longest increasing subsequences
  • Interacting particles systems

Mathematics Subject Classification

  • 60K35
  • 60G55