On the Cycle Structure of Mallows Permutations

TL;DR

This paper analyzes the cycle structure of Mallows permutations, revealing two regimes of cycle sizes depending on the parameter q, and shows that in one regime, cycle lengths follow the Poisson-Dirichlet distribution, similar to uniform permutations.

Contribution

It characterizes the asymptotic cycle length distribution of Mallows permutations and identifies a phase transition between small and macroscopic cycles.

Findings

Expected cycle length scales as min{(1-q)^{-2}, n}

Poisson-Dirichlet distribution for cycle sizes in the macroscopic regime

Existence of a phase transition depending on (1-q)^{-2} relative to n

Abstract

We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $\pi \in \mathbb{S}_n$ is proportional to $q^{\textrm{inv}(\pi)}$ where $0<q\le 1$ and $\textrm{inv}(\pi)$ is the number of inversions in $\pi$. We show that the expected length of the cycle containing a given point is of order $\min\{(1-q)^{-2}, n\}$. This marks the existence of two asymptotic regimes: with high probability, when $n$ tends to infinity with $(1-q)^{-2} \ll n$ then all cycles have size $o(n)$ whereas when $n$ tends to infinity with $(1-q)^{-2}\gg n$ then macroscopic cycles, of size proportional to $n$, emerge. In the second regime, we prove that the distribution of normalized cycle lengths follows the Poisson-Dirichlet law, as in a uniformly random permutation. The results bear formal similarity with a conjectured localization transition for random band matrices. Further results are presented for the variance of the cycle lengths, the expected diameter of cycles and the expected number of cycles. The proofs rely on the exact sampling algorithm for the Mallows distribution and make use of a special diagonal exposure process for the graph of the permutation.