Permutons, meanders, and SLE-decorated Liouville quantum gravity

TL;DR

The paper constructs and analyzes a broad class of random permutons formed from an LQG surface decorated by a pair of space-filling SLE curves, covering the meandric and skew Brownian instances. It proves that any permutation sequence converging to such a permuton has LIS and LDS sublinear in length, and it establishes that the permutons’ closed supports have Hausdorff dimension one, via LQG metric estimates. It also proves a re-rooting invariance for the meandric permuton and derives a pattern-density formula in terms of SLE hitting probabilities and LQG correlation functions, with explicit forms in special cases. The results link combinatorial permutation limits to the physics-inspired framework of SLE-decorated Liouville quantum gravity, and they raise open questions about scaling limits of meanders and deeper pattern densities. Overall, the work provides a rigorous bridge between random permutations, SLE/LQG, and their geometric/dimensional properties, along with several compelling conjectures and directions for future study.

Abstract

We study a class of random permutons which can be constructed from a pair of space-filling Schramm-Loewner evolution (SLE) curves on a Liouville quantum gravity (LQG) surface. This class includes the skew Brownian permutons introduced by Borga (2021), which describe the scaling limit of various types of random pattern-avoiding permutations. Another interesting permuton in our class is the meandric permuton, which corresponds to two independent SLE$_8$ curves on a $γ$-LQG surface with $γ= \sqrt{\frac13 \left( 17 - \sqrt{145} \right)}$. Building on work by Di Francesco, Golinelli, and Guitter (2000), we conjecture that the meandric permuton describes the scaling limit of uniform meandric permutations, i.e., the permutations induced by a simple loop in the plane which crosses a line a specified number of times. We show that for any sequence of random permutations which converges to one of the above random permutons, the length of the longest increasing subsequence is sublinear. This proves that the length of the longest increasing subsequence is sublinear for Baxter, strong-Baxter, and semi-Baxter permutations and leads to the conjecture that the same is true for meandric permutations. We also prove that the closed support of each of the random permutons in our class has Hausdorff dimension one. Finally, we prove a re-rooting invariance property for the meandric permuton and write down a formula for its expected pattern densities in terms of LQG correlation functions (which are known explicitly) and the probability that an SLE$_8$ hits a given set of points in numerical order (which is not known explicitly). We conclude with a list of open problems.

Figures (11)

• Figure 1: The permuton ${\boldsymbol{\pi}}_\sigma$ associated with the permutation $\sigma=1,8,3,2,4,6,7,5$ is equal to 8 times the Lebesgue measure on the gray subset of $[0,1]^2$.
• Figure 2: Venn diagram of the different classes of random permutons considered in this paper. The outer grey oval represents all random permutons of the type constructed in \ref{['eqn-permuton-def']}. The meandric permuton and the skew Brownian permutons are special cases of this construction. In particular, the meandric permuton falls in the subclass of permutons constructed from independent SLEs.
• Figure 3: Left: A meander of size $6$, with the intersection points of the loop and the line labeled by the order in which they are hit by the loop (in black) and the line (in red) starting at the blue vertex. The associated meandric permutation is $\sigma_{\ell}=1,4,3,2,5,12,7,8,9,10,11,6$ (in one-line notation); see Definition \ref{['def-meandric-perm']} below. Right: The same meander of size $6$ seen as a pair of Jordan curves in the sphere.
• Figure 4: Left: Two large uniform meanders $\ell$ of size 256 and 2048. Right: The plots of the two corresponding meandric permutations $\sigma_{\ell}$; the blue dots correspond to the values $(i,\sigma_\ell(i))_{i = 1,\dots,|\sigma_\ell|}$. These simulations are obtained using the Markov chain Monte Carlo algorithm from heitsch2011meander.
• Figure 5: Left: A meander $\ell_n$ and the corresponding meandric permutation $\sigma_{n}=1,4,3,2,5,12,7,8,9,10,11,6$. Right: The meander $\ell_n^{(6)}$ obtained by re-rooting $\ell_n$ at the point $x_{6}$ and the corresponding meandric permutation $\sigma^{(6)}_{n}=1,8,9,10,11,12,7,2,5,4,3,6$.

Theorems & Definitions (69)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Conjecture 1.4
• Remark 1.5
• Definition 1.6
• Definition 1.7
• Conjecture 1.8
• Conjecture 1.9
• Definition 1.10