The $p$-$θ$ relation in mating of trees

TL;DR

This work provides an exact p–θ relation in the mating-of-trees framework linking two Brownian-motions descriptions of SLE/LQG couplings, via LCFT boundary structure constants and imaginary-geometry tools. The authors derive the formula for p/(1−p) in terms of θ and γ, and express the associated involution rate and skew Brownian permuton characteristics, including the inversion-rate statistic E[occ](21,μ) in terms of θ. The methodology hinges on conformal welding of quantum wedges and quantum triangles, with LCFT providing exact solvable inputs; the contour decomposition then translates geometric crossing data into explicit analytic formulas. These results bridge continuum LCFT/SLE descriptions and discrete decorated planar maps, enabling precise analysis of skew Brownian permutons and related models through exact probabilistic and geometric inputs. The findings offer a principled path to compute permutation statistics and limit measures in models arising from bipolar orientations and related map-decorations.

Abstract

In the mating-of-trees approach to Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG), it is natural to consider two pairs of correlated Brownian motions coupled together. This arises in the scaling limit of bipolar-orientation-decorated planar maps (Gwynne-Holden-Sun, 2016) and in the related skew Brownian permuton studied by Borga et al. There are two parameters that can be used to index the coupling between the two pairs of Brownian motions, denoted as $p$ and $θ$ in the literature: $p$ describes the Brownian motions, whereas $θ$ describes the SLE curves on LQG surfaces. In this paper, we derive an exact relation between the two parameters and demonstrate its application to computing statistics of the skew Brownian permuton. Our derivation relies on the synergy between mating-of-trees and Liouville conformal field theory (LCFT), where the boundary three-point function in LCFT provide the exact solvable inputs.

Figures (7)

• Figure 1: An illustration of the mating-of-trees setup. Here $p_\gamma(\theta)$ is the same as the probability where $\eta'(t)$ is on the left of the flow line $\eta_\theta$, and the local time of $|X|$ at 0 is $2c_\gamma(\theta)$ times the quantum length of $\eta'([0,t])\cap\eta_\theta$.
• Figure 2: A quantum triangle with $W_2>\frac{\gamma^2}{2}$ and $W_1,W_3<\frac{\gamma^2}{2}$ embedded as $(D,\phi,a_1,a_2,a_3).$ The two thin disks (colored green) are concatenated with the thick triangle (colored yellow) at points $\tilde{a}_1$ and $\tilde{a}_3$.
• Figure 3: An illustration of Theorem \ref{['thm:weld-0']} for $W<\frac{\gamma^2}{2}$ (left) and $W>\frac{\gamma^2}{2}$ (right). The point $a_2$ is located at the infinite end of the quantum wedge and is not shown in the figure. Note that $2-W>\frac{\gamma^2}{2}$ since $W\in(0,2-\frac{\gamma^2}{2})$.
• Figure 4: Left: Illustration of Theorem \ref{['thm:disk-welding']} with $W_1\ge\frac{\gamma^2}{2}$ and $W_2<\frac{\gamma^2}{2}$. Right: Illustration of Theorem \ref{['thm:disk+QT']} with $W,W_3\ge\frac{\gamma^2}{2}$ and $W_1,W_2<\frac{\gamma^2}{2}$.
• Figure 5: An illustration of Proposition \ref{['prop:disk+QT-0']} with $W_0, W > \frac{\gamma^2}{2}$ (left) and $W_0,W<\frac{\gamma^2}{2}$ (right).

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Definition 2.1: Thick quantum disk
• Definition 2.2: Thick quantum wedge
• Definition 2.3: Thin quantum disk
• Definition 2.4: Thin quantum wedge
• Definition 2.5: Quantum cone