Electronic Journal of Probability

Local limit of the fixed point forest

Tobias Johnson, Anne Schilling, and Erik Slivken

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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

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

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

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Zentralblatt MATH identifier

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

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

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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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