On the cycle structure of Mallows permutations

Alexey Gladkich, Ron Peled

Research output: Contribution to journalArticlepeer-review

Abstract

We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation π ∈ S[double-struck]n is proportional to qinv(π) where q > 0 and inv(π) is the number of inversions in π. 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«n then all cycles have size o(n) whereas when n tends to infinity with (1 - q) -2 » 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.

Original languageEnglish
Pages (from-to)1114-1169
Number of pages56
JournalAnnals of Probability
Volume46
Issue number2
DOIs
StatePublished - 1 Mar 2018

Keywords

  • Cycle structure
  • Delocalization
  • Localization
  • Macroscopic cycles
  • Mallows permutations
  • Phase transition
  • Poisson-Dirichlet law
  • Random band matrices

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