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Cycle structure of random standardized permutations

Aurélien Guerder

Abstract

In this article, we study a model of random permutations, which we call random standardized permutations, based on a sequence of i.i.d. random variables. This model generalizes others, such as the riffle-shuffle and the major-index-biased permutations. We first establish an exact result on the joint distribution of the number of cycles of given lengths, involving the notion of primitive words. From this result, we obtain various convergence results, most of which are proved using the method of moments. First we prove that the number of small cycles may have either a Poisson limit distribution, or a limit distribution given by a countable sum of independent geometric distributions. Then we establish a limit distribution for large cycles, which is the Poisson-Dirichlet process. Finally we prove a central limit theorem for the total number of cycles.

Cycle structure of random standardized permutations

Abstract

In this article, we study a model of random permutations, which we call random standardized permutations, based on a sequence of i.i.d. random variables. This model generalizes others, such as the riffle-shuffle and the major-index-biased permutations. We first establish an exact result on the joint distribution of the number of cycles of given lengths, involving the notion of primitive words. From this result, we obtain various convergence results, most of which are proved using the method of moments. First we prove that the number of small cycles may have either a Poisson limit distribution, or a limit distribution given by a countable sum of independent geometric distributions. Then we establish a limit distribution for large cycles, which is the Poisson-Dirichlet process. Finally we prove a central limit theorem for the total number of cycles.
Paper Structure (27 sections, 49 theorems, 141 equations, 7 figures)

This paper contains 27 sections, 49 theorems, 141 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background and model
  3. Overview of the results
  4. Exact distribution of cycles
  5. Asymptotic behavior of short cycles
  6. Asymptotic behavior of large cycles
  7. Asymptotic behavior of cycle count
  8. Random standardized permutations model
  9. Definition
  10. Some particular cases
  11. Atomless variables and uniform model
  12. Major-index-biased permutations
  13. Riffle-Shuffle
  14. Runs of a standardized permutation
  15. Exact distribution of cycle count
  16. ...and 12 more sections

Key Result

Theorem 1.1

For any $k_1, \dots ,k_r \geq 1, \mathbf{i}_1 \in Q_{k_1}, \dots , \mathbf{i}_r \in Q_{k_r}$ pairwise non conjugate, and $l_1, \dots ,l_r \geq 0$,

Figures (7)

  • Figure 1: Standardized permutation of $g = (6,1,5,3,3,1,2)$ and of a sequence of length 50 taking values in $\llbracket 1,6 \rrbracket$, where the colors denote the points whose x-coordinates are $k$ such that $g_k = 1$, then 2, up to $6$ (from bottom to top). Note that we plot the $i$ on the horizontal axis and the $\sigma(i)$ on the vertical axis
  • Figure 2: Example of a major-index-biased permutation with $n = 1000$ and $q = 0.7$
  • Figure 3: A 2-riffle-shuffle of a deck of 52 cards
  • Figure 4: Example on runs with $g = (6,1,5,3,3,1,2)$ and with the sequence of Figure \ref{['IllustrAlgo']}
  • Figure 5: Example with $g = (1,6,5,3,3,1,2)$, here $D_1 = 1$ (and 1 is a fixed point of type $1$ of $\mathrm{std}(g)$, and $D_3 = 2$ (and 4 and 5 are fixed points of type $3$ of $\mathrm{std}(g)$).
  • ...and 2 more figures

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • ...and 81 more