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March 2018 On the cycle structure of Mallows permutations
Alexey Gladkich, Ron Peled
Ann. Probab. 46(2): 1114-1169 (March 2018). DOI: 10.1214/17-AOP1202


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.


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Alexey Gladkich. Ron Peled. "On the cycle structure of Mallows permutations." Ann. Probab. 46 (2) 1114 - 1169, March 2018.


Received: 1 January 2016; Revised: 1 May 2017; Published: March 2018
First available in Project Euclid: 9 March 2018

zbMATH: 06864081
MathSciNet: MR3773382
Digital Object Identifier: 10.1214/17-AOP1202

Primary: 05A05 , 60C05
Secondary: 60B20 , 60B99 , 60F05 , 60K35 , 82B23 , 82B26

Keywords: Cycle structure , delocalization , Localization , macroscopic cycles , Mallows permutations , phase transition , Poisson–Dirichlet law , random band matrices

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 2 • March 2018
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