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The permuton limit of random recursive separable permutations

Valentin Féray, Kelvin Rivera-Lopez

Abstract

We introduce and study a simple Markovian model of random separable permutations. Our first main result is the almost sure convergence of these permutations towards a random limiting object in the sense of permutons, which we call the recursive separable permuton. We then prove several results on this new limiting object: a characterization of its distribution via a fixed-point equation, a combinatorial formula for its expected pattern densities, an explicit integral formula for its intensity measure, and lastly, we prove that its distribution is absolutely singular with respect to that of the Brownian separable permuton, which is the large size limit of uniform random separable permutations.

The permuton limit of random recursive separable permutations

Abstract

We introduce and study a simple Markovian model of random separable permutations. Our first main result is the almost sure convergence of these permutations towards a random limiting object in the sense of permutons, which we call the recursive separable permuton. We then prove several results on this new limiting object: a characterization of its distribution via a fixed-point equation, a combinatorial formula for its expected pattern densities, an explicit integral formula for its intensity measure, and lastly, we prove that its distribution is absolutely singular with respect to that of the Brownian separable permuton, which is the large size limit of uniform random separable permutations.
Paper Structure (30 sections, 31 theorems, 137 equations, 15 figures)

This paper contains 30 sections, 31 theorems, 137 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Our model
  3. The limiting permuton
  4. Properties of the recursive separable permuton
  5. Self-similarity
  6. Expected pattern densities
  7. The intensity measure
  8. Analog results for cographs
  9. Outline of the paper
  10. Background
  11. Permutation patterns.
  12. Wasserstein metric.
  13. Push-forward permutons.
  14. Permutations and pairs of total orders
  15. Construction and convergence
  16. ...and 15 more sections

Key Result

Theorem 1.2

The random permutations $( \sigma^{(n),p} )_{ n \ge 1 }$ converge a.s. to a random permuton. We call this permuton the recursive separable permuton (of parameter $p$) and denote it by $\bm\mu^{\text{\tiny rec}}_p$.

Figures (15)

  • Figure 1: Examples of possible inflation steps from $\tau:=\sigma^{(n),p}$ to $\rho:=\sigma^{(n+1),p}$. For $i=1$ or $2$, the point of $\tau_i$ chosen uniformly at random, as well as the two new adjacent points in $\rho_i$ replacing it, are painted in red. In $\rho_1$, these two new points are in increasing order (we say that we have performed an increasing inflation), while, in $\rho_2$, they are in decreasing order (in this case, we have performed a decreasing inflation).
  • Figure 2: A sample of permutations $\sigma^{(10),{\, 0.5}}$, $\sigma^{(100),{\, 0.5}}$, $\sigma^{(1000),{\, 0.5}}$ corresponding to the same realization of the process $(\sigma^{(n),{\, 0.5}})_{n \ge 1}$.
  • Figure 3: The permuton $\mu \otimes_{(u,S)} \nu$.
  • Figure 4: Direct sum and skew sum of permutations.
  • Figure 5: A rooted increasing binary tree and the associated permutation. The permutation associated to the left subtree of the root is $4132$, the one associated to its right-subtree $1243$. Since the root has decoration $\oplus$, the permutation associated to the whole tree is $4132 \oplus 1243$, which is equal to $41325687$.
  • ...and 10 more figures

Theorems & Definitions (63)

  • Remark 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Remark 1.8
  • Proposition 1.9
  • Corollary 1.10
  • ...and 53 more