Electronic Journal of Probability

Local limit of the fixed point forest

Tobias Johnson, Anne Schilling, and Erik Slivken

Full-text: Open access

Abstract

Consider the following partial “sorting algorithm” on permutations: take the first entry of the permutation in one-line notation and insert it into the position of its own value. Continue until the first entry is $1$. This process imposes a forest structure on the set of all permutations of size $n$, where the roots are the permutations starting with $1$ and the leaves are derangements. Viewing the process in the opposite direction towards the leaves, one picks a fixed point and moves it to the beginning. Despite its simplicity, this “fixed point forest” exhibits a rich structure. In this paper, we consider the fixed point forest in the limit $n\to \infty $ and show using Stein’s method that at a random permutation the local structure weakly converges to a tree defined in terms of independent Poisson point processes. We also show that the distribution of the length of the longest path from a random permutation to a leaf converges to the geometric distribution with mean $e-1$, and the length of the shortest path converges to the Poisson distribution with mean $1$. In addition, the higher moments are bounded and hence the expectations converge as well.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 18, 26 pp.

Dates
Received: 20 July 2016
Accepted: 4 February 2017
First available in Project Euclid: 15 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1487127646

Digital Object Identifier
doi:10.1214/17-EJP36

Mathematical Reviews number (MathSciNet)
MR3622888

Zentralblatt MATH identifier
1357.60092

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 05C05: Trees 05A05: Permutations, words, matrices 60C05: Combinatorial probability 60F99: None of the above, but in this section

Keywords
permutations trees fixed points sorting algorithm Poisson point process Stein’s method weak convergence

Rights
Creative Commons Attribution 4.0 International License.

Citation

Johnson, Tobias; Schilling, Anne; Slivken, Erik. Local limit of the fixed point forest. Electron. J. Probab. 22 (2017), paper no. 18, 26 pp. doi:10.1214/17-EJP36. https://projecteuclid.org/euclid.ejp/1487127646


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