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The skew Brownian permuton: a new universality class for random constrained permutations

This paper introduces the skew Brownian permuton, a new two-parameter family of random permutons constructed via flows of solutions to skew perturbed Tanaka equations driven by correlated Brownian excursions, which unifies the Baxter and biased Brownian separable permutons as special cases and establishes a direct continuum-level connection between constrained permutations and Liouville quantum gravity spheres decorated with Schramm-Loewner evolution curves.

Original authors: Jacopo Borga

Published 2026-09-30

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CC BY 4.0

Original authors: Jacopo Borga

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where you could watch a crowd of people shuffle, swap places, and rearrange themselves according to a set of hidden rules. If you watched a small group, you might see a chaotic jumble. But if you watched a crowd of millions, a pattern might emerge from the chaos, a shape that reveals the underlying logic of their movement. In the field of probability, mathematicians study these patterns by looking at permutations, which are simply different ways of ordering a list of items. For decades, scientists have tried to understand what happens to these lists when they grow infinitely large. Some lists, like those sorted by simple rules, settle into a predictable, smooth shape. Others, which follow more complex constraints, break apart into jagged, fractal forms that look like rough, self-similar coastlines. These fractal shapes are called permutons, and they represent the ultimate limit of how these random lists can arrange themselves.

Until now, scientists had identified a few specific types of these fractal shapes, each arising from a different set of rules. One type appeared when studying permutations that avoid certain specific patterns, while another emerged from a different family of constrained lists. These two types seemed to belong to separate universes, with no obvious connection between them. A researcher at Stanford University, Jacopo Borga, has now built a bridge between these two worlds. He has constructed a new, flexible family of shapes that unifies these previously separate cases into a single, larger category. This new family, which he calls the skew Brownian permuton, acts as a master key, showing that the different fractal shapes observed in random permutations are actually just different settings of the same underlying mechanism.

To build this new family, Borga turned to a tool from the world of physics and random motion: the Brownian excursion. Imagine a particle drifting randomly in a two-dimensional space, but with a strict rule that it must stay within a specific corner of the room and eventually return to the starting point. This path, known as a Brownian excursion, is a jagged, unpredictable line that never leaves its designated area. Borga used these paths as the driving force for a new set of mathematical equations. These equations describe how a collection of other paths, starting at different times, move and interact with the main Brownian path. When these interacting paths hit a boundary, they behave in a specific way, sometimes bouncing back and sometimes crossing over, depending on a set of parameters Borga introduced.

The brilliance of this construction lies in its ability to capture different behaviors by simply adjusting two numbers. One number controls how the paths interact with the boundary of their space, while the other controls the direction they take when they touch that boundary. When Borga set these numbers to specific values, the resulting shape matched the fractal pattern known as the Baxter permuton, which had been discovered in the study of permutations avoiding two specific patterns. When he adjusted the numbers to a different setting, the shape transformed into the biased Brownian separable permuton, a shape previously linked to a different class of constrained lists. By showing that both of these distinct shapes are just special cases of his new family, Borga proved that they are part of the same larger universality class.

The paper does not stop at just connecting these two known shapes. Borga also demonstrated that this new family of shapes appears in a completely different area of mathematics and physics: the study of random surfaces and quantum gravity. He showed that if you take a random, fluctuating surface and draw two specific types of curves across it, the way these curves interact creates a shape that is identical to his skew Brownian permuton. This connection is significant because it links the abstract world of rearranging lists to the physical world of random surfaces and the geometry of space-time. It suggests that the same mathematical laws govern both the ordering of numbers and the structure of the universe at a quantum level.

To ensure that this new family of shapes was mathematically sound, Borga had to prove that the equations he used actually had solutions and that those solutions were unique. This was a difficult task because the equations involved a type of randomness that can be tricky to handle, especially when the paths are forced to stay within a specific region. He developed new mathematical techniques to show that for almost every setting of his two parameters, a unique, well-defined shape exists. He also ran computer simulations to visualize these shapes, generating images that show how the fractal patterns change as the parameters shift. These simulations revealed a rich variety of forms, ranging from dense, clustered shapes to more spread-out, scattered patterns, all generated by the same underlying process.

The work also opens the door to understanding other types of constrained permutations. Borga identified a specific model of random lists, called strong-Baxter permutations, that converges to a version of his new shape with parameters that do not match any previously known case. This suggests that there are likely many more types of random permutations that fit into this new, larger family, waiting to be discovered. The paper concludes by listing several questions that remain unanswered, such as the exact dimensions of these fractal shapes and how often certain patterns appear within them. These questions point toward future research that could further refine our understanding of how randomness and constraint interact to create order.

In essence, this paper provides a new map for a territory that was previously thought to be divided into separate islands. By constructing a single, continuous family of shapes, Borga has shown that the diverse behaviors of random constrained permutations are not isolated phenomena but are instead variations of a single, fundamental process. This unification offers a deeper understanding of the limits of randomness and provides powerful new tools for mathematicians and physicists to explore the hidden structures that govern complex systems. The discovery does not just add a new shape to the collection; it changes the way we see the entire landscape of random permutations, revealing a unity that was previously invisible.

Technical Summary: The Skew Brownian Permuton

Problem Statement
The study of limits of random permutations has evolved from analyzing specific statistics (e.g., inversions, longest increasing subsequences) to a geometric approach known as permuton limits, where the permutation itself converges to a random measure on the unit square. While various models of constrained permutations have been shown to converge to specific random fractal permutons—most notably the Brownian separable permuton (limit of separable permutations) and the Baxter permuton (limit of Baxter permutations)—these objects were previously understood as distinct universality classes. The central problem addressed in this paper is to determine the relationship between these limiting objects and to construct a unified framework that encompasses them, as well as other emerging models of constrained permutations.

Methodology
The paper introduces a new family of random permutons, termed the skew Brownian permuton, constructed via a continuous coalescent-walk process driven by stochastic differential equations (SDEs).

Construction via SDEs: The core of the construction involves a family of SDEs indexed by $u \in [0, 1]$ and driven by a two-dimensional correlated Brownian excursion $E_\rho = (X_\rho, Y_\rho)$ in the non-negative quadrant. The equations, referred to as skew perturbed Tanaka equations, are given by:
$dZ^{(u)}_{\rho,q}(t) = \mathbb{1}_{\{Z^{(u)}_{\rho,q}(t)>0\}} dY_\rho(t) - \mathbb{1}_{\{Z^{(u)}_{\rho,q}(t)\le0\}} dX_\rho(t) + (2q - 1) \cdot dL^{Z^{(u)}_{\rho,q}}_t$
where $L$ denotes the symmetric local time at zero, $\rho \in (-1, 1]$ is the correlation parameter, and $q \in [0, 1]$ is a skewness parameter.
Existence and Uniqueness: The author proves the existence and pathwise uniqueness of strong solutions for these SDEs. For $\rho \in (-1, 1)$ , this is achieved by establishing pathwise uniqueness for the corresponding equations driven by a correlated Brownian motion (using absolute continuity arguments and results from the literature on perturbed Tanaka equations) and transferring these results to the excursion case. For the degenerate case $\rho = 1$ , where the driving process reduces to a one-dimensional Brownian excursion, the author utilizes a construction based on stochastic flows of maps (inspired by Hajri and Le Jan–Raimond) to define solutions that are not measurable functions of the driving process alone but require additional randomness.
Permuton Definition: From the collection of processes $\{Z^{(u)}_{\rho,q}\}_{u \in [0,1]}$ , a stochastic process $\phi_{Z_{\rho,q}}$ is defined, representing the "continuum permutation." The skew Brownian permuton $\mu_{\rho,q}$ is then defined as the push-forward of the Lebesgue measure via the mapping $(\text{Id}, \phi_{Z_{\rho,q}})$ .

Key Contributions and Results

Unification of Universality Classes: The primary contribution is the identification of the skew Brownian permuton as a new, larger universality class. The paper proves that:
- The Baxter permuton corresponds to the specific case where $\rho = -1/2$ and $q = 1/2$ .
- The biased Brownian separable permuton (the limit of uniform permutations in substitution-closed classes) corresponds to the case where $\rho = 1$ and $q = 1-p$ (where $p$ is the bias parameter).
  This unifies two previously distinct limiting objects under a single parametric family.
Convergence of Discrete Models: The paper establishes that the skew Brownian permuton arises as the limit of several specific families of random constrained permutations. Notably, it cites companion work [Bor22] showing that:
- Strong-Baxter permutations converge to a skew Brownian permuton with non-degenerate parameters ( $\rho \approx -0.2151, q \approx 0.3008$ ).
- Semi-Baxter permutations converge to a skew Brownian permuton with $\rho = (-1+\sqrt{5})/4$ and $q = 1/2$ .
  This demonstrates that the family covers a range of parameters not previously realized by known permutation models.
Connection to Liouville Quantum Gravity (LQG) and SLE: The paper establishes a direct connection between the skew Brownian permuton and the geometry of Liouville quantum gravity spheres decorated with two coupled Schramm-Loewner evolution (SLE) curves. Specifically, it proves that the permuton induced by the interaction of two space-filling SLE curves on a $\gamma$ -LQG sphere is a skew Brownian permuton. The parameters $\rho$ and $q$ are linked to the LQG parameter $\gamma$ and the SLE angle $\theta$ . This result bridges discrete combinatorial models (constrained permutations) and continuum random geometry (decorated planar maps) at the continuum level.
Mathematical Rigor: The paper provides rigorous proofs for the well-definedness of the skew Brownian permuton for all admissible parameters $(\rho, q) \in (-1, 1] \times [0, 1]$ , including the technical handling of the $\rho=1$ case where pathwise uniqueness fails in the standard sense.

Significance
The paper claims to identify a new universality class that brings together the Baxter permuton and the biased Brownian separable permuton, explaining their relationship through a unified construction. By linking these combinatorial limits to SDEs driven by correlated Brownian excursions and to SLE-decorated LQG spheres, the work provides a powerful new toolkit for analyzing random constrained permutations. The author suggests that this connection allows for the transfer of convergence results between discrete permutation models and discrete planar map models. Furthermore, the paper offers a new SDE interpretation for quantities related to SLE-decorated LQG spheres, a connection not explicitly exploited in prior Liouville quantum gravity literature.

Open Problems
The paper concludes by outlining several open questions, including:

Determining the intensity measure and Hausdorff dimension of the skew Brownian permuton.
Analyzing the properties of the underlying stochastic process $\phi_{Z_{\rho,q}}$ .
Computing the limiting distribution of pattern occurrences within the permuton.
Investigating the length of the longest increasing subsequence in permutations converging to this limit.
Extending the construction to the case $\rho = -1$ and establishing the connection between the biased Brownian separable permuton and critical 2-LQG spheres.

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