From Hammersley's lines to Hammersley's trees

• 成果类型: Article
• 署名作者: Basdevant, A. -L.; Gerin, L.; Gouere, J. -B.; Singh, A.
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; Universite de Tours; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0772-2
• 发表日期: 2018
• 页码: 1-51
• 关键词: longest increasing subsequences; sinks

摘要

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.