Local limit of the fixed point forest

TL;DR

This paper studies the local structure of the fixed point forest on permutations as size grows infinitely large, revealing convergence to a tree structure with specific probabilistic distributions for path lengths.

Contribution

It establishes the weak convergence of local permutation structures to a Poisson-based tree and characterizes the limiting distributions of path lengths.

Findings

Local structure converges to a tree defined by Poisson point processes.

Longest path length converges to a geometric distribution with mean e-1.

Shortest path length converges to a Poisson distribution with mean 1.

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 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.