From Hammersley's lines to Hammersley's trees

TL;DR

This paper introduces Hammersley's tree process, a stationary random tree model extending Hammersley's line process, which counts heaps needed for random permutations and grows logarithmically with permutation size.

Contribution

It constructs a new stationary random tree model with prescribed offspring distribution, extending Hammersley's process and linking it to heap counting in permutations.

Findings

Number of heaps grows logarithmically with permutation size

Model extends Hammersley's line process to trees

Provides a combinatorial interpretation for the process

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 (2011) and Istrate and Bonchis (2015) in the case of regular trees. We show, in particular, that the number of heaps grows logarithmically with the size of the permutation.