Open Access
2017 Local limit of the fixed point forest
Tobias Johnson, Anne Schilling, Erik Slivken
Electron. J. Probab. 22: 1-26 (2017). DOI: 10.1214/17-EJP36


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.


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Tobias Johnson. Anne Schilling. Erik Slivken. "Local limit of the fixed point forest." Electron. J. Probab. 22 1 - 26, 2017.


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

zbMATH: 1357.60092
MathSciNet: MR3622888
Digital Object Identifier: 10.1214/17-EJP36

Primary: 60J80
Secondary: 05A05 , 05C05 , 60C05 , 60F99

Keywords: Fixed points , permutations , Poisson point process , sorting algorithm , Stein’s method , trees , weak convergence

Vol.22 • 2017
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