Local limit of the fixed point forest

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.