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High-dimensional permutons: theory and applications

Jacopo Borga, Andrew Lin

TL;DR

The paper extends permuton theory to d dimensions by defining d-permutons and establishing a framework in which convergence is equivalent to the convergence of all pattern frequencies. It then identifies two natural high-dimensional random limits: the 3D Schnyder wood permuton and the d-dimensional Brownian separable permuton, each described via connections to Brownian excursions, coalescent-walk processes, and universal 2D permutons, with links to CRT, SLE, and LQG. The Schnyder wood result provides a random 3D permuton limit μ_S governed by coupled skew Brownian permutons, while the d-separable result shows convergence to μ^B_{1/2,...,1/2}, not determined by lower-dimensional marginals. Taken together, these results illustrate a universal high-dimensional permuton landscape and establish a robust probabilistic-combinatorial approach for analyzing large-scale d-permutations with potential further universality across natural higher-dimensional permutation classes.

Abstract

Permutons, which are probability measures on the unit square $[0, 1]^2$ with uniform marginals, are the natural scaling limits for sequences of (random) permutations. We introduce a $d$-dimensional generalization of these measures for all $d \ge 2$, which we call $d$-dimensional permutons, and extend -- from the two-dimensional setting -- the theory to prove convergence of sequences of (random) $d$-dimensional permutations to (random) $d$-dimensional permutons. Building on this new theory, we determine the random high-dimensional permuton limits for two natural families of high-dimensional permutations. First, we determine the $3$-dimensional permuton limit for Schnyder wood permutations, which bijectively encode planar triangulations decorated by triples of spanning trees known as Schnyder woods. Second, we identify the $d$-dimensional permuton limit for $d$-separable permutations, a pattern-avoiding class of $d$-dimensional permutations generalizing ordinary separable permutations. Both high-dimensional permuton limits are random and connected to previously studied universal 2-dimensional permutons, such as the Brownian separable permutons and the skew Brownian permutons, and share interesting connections with objects arising from random geometry, including the continuum random tree, Schramm--Loewner evolutions, and Liouville quantum gravity surfaces.

High-dimensional permutons: theory and applications

TL;DR

The paper extends permuton theory to d dimensions by defining d-permutons and establishing a framework in which convergence is equivalent to the convergence of all pattern frequencies. It then identifies two natural high-dimensional random limits: the 3D Schnyder wood permuton and the d-dimensional Brownian separable permuton, each described via connections to Brownian excursions, coalescent-walk processes, and universal 2D permutons, with links to CRT, SLE, and LQG. The Schnyder wood result provides a random 3D permuton limit μ_S governed by coupled skew Brownian permutons, while the d-separable result shows convergence to μ^B_{1/2,...,1/2}, not determined by lower-dimensional marginals. Taken together, these results illustrate a universal high-dimensional permuton landscape and establish a robust probabilistic-combinatorial approach for analyzing large-scale d-permutations with potential further universality across natural higher-dimensional permutation classes.

Abstract

Permutons, which are probability measures on the unit square with uniform marginals, are the natural scaling limits for sequences of (random) permutations. We introduce a -dimensional generalization of these measures for all , which we call -dimensional permutons, and extend -- from the two-dimensional setting -- the theory to prove convergence of sequences of (random) -dimensional permutations to (random) -dimensional permutons. Building on this new theory, we determine the random high-dimensional permuton limits for two natural families of high-dimensional permutations. First, we determine the -dimensional permuton limit for Schnyder wood permutations, which bijectively encode planar triangulations decorated by triples of spanning trees known as Schnyder woods. Second, we identify the -dimensional permuton limit for -separable permutations, a pattern-avoiding class of -dimensional permutations generalizing ordinary separable permutations. Both high-dimensional permuton limits are random and connected to previously studied universal 2-dimensional permutons, such as the Brownian separable permutons and the skew Brownian permutons, and share interesting connections with objects arising from random geometry, including the continuum random tree, Schramm--Loewner evolutions, and Liouville quantum gravity surfaces.
Paper Structure (24 sections, 42 theorems, 169 equations, 15 figures)

This paper contains 24 sections, 42 theorems, 169 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Characterization of high-dimensional permuton convergence
  3. Random high-dimensional permuton limits for Schnyder wood permutations and $d$-separable permutations
  4. The 3-dimensional Schnyder wood permuton
  5. The $d$-dimensional Brownian separable permuton
  6. Outline, key ideas for the proofs, and final remarks
  7. Acknowledgments.
  8. High-dimensional permuton theory
  9. The theory for the deterministic setting
  10. The theory for the random setting
  11. Background: the skew Brownian permutons and coalescent-walk processes
  12. The skew Brownian permutons
  13. The Brownian separable $d$-permutons
  14. Coalescent-walk processes
  15. The random 3-dimensional permuton limit of Schnyder wood permutations
  16. ...and 9 more sections

Key Result

Theorem 1.5

Let $\sigma_n$ be a random $d$-dimensional permutation of size $n$ for all $n \in \mathbb N$. Then the following are equivalent: When these conditions hold, we have for all $d$-dimensional permutations $\tau$ of size $k$ that and furthermore the vector $(v_\tau)_\tau$ is identically distributed as $(\text{freq}(\tau, \mu))_\tau$ and $\rho_k=P_\mu[k]$ in distribution.

Figures (15)

  • Figure 1: Simulations for two 3-dimensional permutons with their respective 2-dimensional marginal permutons. Left: The 3-dimensional permuton associated with a permutation of size 10000 sampled from the Schnyder wood permuton of \ref{['mainresult3']}. Right: The 3-dimensional permuton associated with a permutation of size 10000 sampled from the Brownian separable $3$-permuton of \ref{['mainresult4']}. Animated simulations can be found at http://www.jacopoborga.com/2024/12/27/high-dimensional-permutons-the-schnyder-wood-and-brownian-separable-d-permuton/.
  • Figure 2: In all images, the colors are meant only for visual aid (with boxes colored from red to blue as the $x$-coordinate ranges from $0$ to $1$). Left: The (2-dimensional) permuton associated to the permutation $\sigma = (3, 2, 5, 1, 4)$ (as defined in \ref{['eq:permut-perm']}). Each shaded square (of side length $\frac{1}{5}$) is uniformly assigned a total probability mass of $\frac{1}{5}$. Middle-right: The 3-dimensional permuton associated to the $3$-dimensional permutation $\sigma$ of size $5$ (as introduced in \ref{['eq:defn-d-perm']}) defined by $\sigma(1) = (1, 3), \sigma(2) = (5, 2), \sigma(3) = (2, 5), \sigma(4) = (3, 1), \sigma(5) = (4, 4)$, shown from two different angles. In shorthand, following \ref{['eq:perm-shorthand']}, this permutation may also be written as $((1,5,2,3,4), (3,2,5,1,4))$. Notice that the left $2$-permuton is a marginal of the right $3$-permuton (specifically the projection onto the coordinates $(X, Z)$).
  • Figure 3: A Schnyder wood triangulation of size 10 with its corresponding Schnyder wood permutation on top. The roots are colored in their respective colors. We include only the blue labels for clarity. This Schnyder wood will be used as a running example also for several constructions in \ref{['schnydersection']}.
  • Figure 4: The permutons associated to a uniformly sampled Schnyder wood permutation of size $200$ (left) and a uniformly sampled $3$-separable permutation of size $40077$ (right). Much like in \ref{['fig:permuton']}, the color indicates the $x$-coordinate of the box (ranging from red to blue as $x$ ranges from $0$ to $1$). The two-dimensional marginals (projecting down onto the first and second coordinates and onto the first and third coordinates) are shown below their corresponding permutons. We highlight that the main difference with \ref{['fig:3dpermutons']} is that here the two simulations are for uniform Schnyder wood and 3-separable permutations, respectively, while in \ref{['fig:3dpermutons']} the simulations are obtained by sampling permutations from the corresponding limiting permutons, as discovered in \ref{['mainresult3']} and \ref{['mainresult4']}.
  • Figure 5: Two Schnyder woods $M_1$ and $M_2$ of size $3$, where $\sigma_{M_1}^g = \sigma_{M_2}^g$ but $\sigma_{M_1}^r \ne \sigma_{M_2}^r$.
  • ...and 10 more figures

Theorems & Definitions (114)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Theorem 1.9
  • Proposition 1.10
  • ...and 104 more