The Annals of Probability

On the cycle structure of Mallows permutations

Alexey Gladkich and Ron Peled

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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 $q^{\operatorname{inv}(\pi)}$ where $q>0$ and $\operatorname{inv}(\pi)$ is the number of inversions in $\pi$.

We focus on the case that $q<1$ and 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.

Article information

Ann. Probab., Volume 46, Number 2 (2018), 1114-1169.

Received: January 2016
Revised: May 2017
First available in Project Euclid: 9 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05A05: Permutations, words, matrices
Secondary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B23: Exactly solvable models; Bethe ansatz 82B26: Phase transitions (general) 60B99: None of the above, but in this section 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Mallows permutations cycle structure Poisson–Dirichlet law phase transition macroscopic cycles localization delocalization random band matrices


Gladkich, Alexey; Peled, Ron. On the cycle structure of Mallows permutations. Ann. Probab. 46 (2018), no. 2, 1114--1169. doi:10.1214/17-AOP1202.

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