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Permuton limit of a generalization of the Mallows and $k$-card-minimum models

Joanna Jasińska, Balázs Ráth

TL;DR

The paper develops a unifying framework for random permutation models via the general $PERM(g,n)$ construction, encompassing Mallows and $k$-card-minimum models. It proves a permuton-level law of large numbers: the sequence $\sigma_n$ converges in probability to a deterministic permuton $\mu_g$ with c.d.f. $F_g$ defined through $V_g$, yielding convergence of pattern densities to their permuton counterparts. A universal diagonal-band phenomenon is established: under a high-parameter limit, the mass concentrates near the diagonal and conditional on a horizontal coordinate, the vertical displacement converges to a logistic distribution, linking Mallows and $k$CM through a common limiting behavior. The framework also ensures symmetry properties, including inversion invariance and $F_g$-symmetry, strengthening the structural parallels between these models and validating Travers’ universality conjecture for the band structure.

Abstract

We introduce and study a new random permutation model that generalizes the $k$-card minimum model defined by Travers and the Mallows model. We calculate the permuton limit of such a sequence of random permutations. As a corollary, we deduce the law of large numbers for pattern densities. Moreover, we prove a universality result about the band structure of the limiting permuton, confirming a conjecture of Travers about the $k$-card minimum model. More specifically, we show that if a certain model parameter goes to infinity then the appropriately scaled restriction of the permuton measure to a line that intersects the diagonal perpendicularly converges weakly to the logistic distribution.

Permuton limit of a generalization of the Mallows and $k$-card-minimum models

TL;DR

The paper develops a unifying framework for random permutation models via the general construction, encompassing Mallows and -card-minimum models. It proves a permuton-level law of large numbers: the sequence converges in probability to a deterministic permuton with c.d.f. defined through , yielding convergence of pattern densities to their permuton counterparts. A universal diagonal-band phenomenon is established: under a high-parameter limit, the mass concentrates near the diagonal and conditional on a horizontal coordinate, the vertical displacement converges to a logistic distribution, linking Mallows and CM through a common limiting behavior. The framework also ensures symmetry properties, including inversion invariance and -symmetry, strengthening the structural parallels between these models and validating Travers’ universality conjecture for the band structure.

Abstract

We introduce and study a new random permutation model that generalizes the -card minimum model defined by Travers and the Mallows model. We calculate the permuton limit of such a sequence of random permutations. As a corollary, we deduce the law of large numbers for pattern densities. Moreover, we prove a universality result about the band structure of the limiting permuton, confirming a conjecture of Travers about the -card minimum model. More specifically, we show that if a certain model parameter goes to infinity then the appropriately scaled restriction of the permuton measure to a line that intersects the diagonal perpendicularly converges weakly to the logistic distribution.

Paper Structure

This paper contains 5 sections, 15 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statements of main results
  3. Law of large numbers in permuton space
  4. Band structure of the limiting permuton
  5. Symmetry

Key Result

Corollary 2.3

For any $\beta \in \mathbb{R}$ we recover the $(n,q)$-Mallows model with parameter $q=e^{-\frac{\beta}{n}}$ as a special case of Definition def_gen_perm_model if we choose

Figures (1)

  • Figure 1: The image on the left is the matrix of a random permutation generated according to the $k$CM with parameters $n=10^4$ and $k=20$. The image on the right is the matrix of a random permutation generated according to the $(n,q)$-Mallows model with parameters $n=10^4$ and $q=e^{\frac{-\beta}{n}}$, where $\beta=20$. Note the visible difference: $\psi(s)=\frac{1}{1-s}$ on the left, while $\psi(s)\equiv 1$ on the right, cf. Remark \ref{['remark_kcm_mallows_satisfy_assump']}.

Theorems & Definitions (36)

  • Definition 2.1: $g$-random permutation
  • Definition 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Definition 2.5: Permuton
  • Definition 2.6: Permutations as permutons
  • Definition 2.7: Permuton arising from $g$
  • Theorem 2.8: Law of large numbers in permuton space
  • Corollary 2.9: Mallows and $k$CM limiting permutons
  • Remark 2.10: Symmetry
  • ...and 26 more