On the cycle structure of Mallows permutations

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 n is proportional to q inv(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. PermutonsThe regime of parameters in which n · (1 − q) → β is also of special interest as in this case there is a limiting density to the empirical measure of the points in the graph of a Mallows permutation. Starr [27] obtained an explicit formula for the limiting density as a function of β. In modern terminology, the limiting density is called a permuton. Recently, Mukherjee [22] proved Poisson limit theorems for the lengths of short cycles for models converging to permutons, including the Mallows model as a special case. See also Kenyon, Král', Radin and Winkler [17] for relations with permutons with fixed pattern densities.

Regenerative random permutations of integers

“…The long-standing conjectured central limit theorem was solved by Baik, Deift and Johansson [8]. Recently, limit theorems for large Mallows permutations have been considered by Mueller and Starr [77], Bhatnagar and Peled [13], Basu and Bhatnagar [9], Gladkich and Peled [38]. (ii).…”

Finite Automata, Probabilistic Method, and Occurrence Enumeration of a Pattern in Words and Permutations

“…Having in hand the above estimates for E(A n ) we can now evaluate the variance of X n . Taking into the account (17), (19), and (20), we obtain that…”

“…Both the regimes can be considered as a perturbation of a uniform distribution, over S n in the former case and over the pattern-avoiding set {w ∈ [k] n : occ v (w) = 0} in the latter. In the context of permutations, similar regimes for the particular case when the pattern is the inversion 21, were recently studied in [6,19,33].…”

Limit Distributions for Euclidean Random Permutations

“…We mention that the results regarding the emergence of macroscopic cycles in one dimension bear formal similarity with a conjectured localization transition for random band matrices. This similarity is detailed in [26,Section 1.2.2] in the context of the Mallows measure on permutations. One may also define random band matrices in dimensions d ≥ 2 where they are rather poorly understood.…”