[Paper Review] Conformal welding of quantum disks

Morris Ang, Nina Holden+1 more|arXiv (Cornell University)|Sep 17, 2020

Stochastic processes and statistical mechanics36 references4 citations

TL;DR

This paper establishes the conformal welding of quantum disks in Liouville quantum gravity, proving that welding two quantum disks of weights $W_1, W_2 > 0$ along their boundary lengths yields a new quantum disk decorated with an independent chordal $\operatorname{SLE}_{\kappa}(\rho_{-};\rho_{+})$ curve, where $\kappa = \gamma^2$. The result extends the classical welding of quantum wedges to finite-area surfaces and provides a unified framework for mating-of-trees theorems on the quantum disk and sphere.

ABSTRACT

Two-pointed quantum disks with a weight parameter $W > 0$ are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their boundary lengths gives another quantum disk decorated with an independent chordal $\mathrm{SLE}_κ(ρ_-;ρ_+)$ curve. This is the finite-volume counterpart of the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces called quantum wedges, which is fundamental to the mating-of-trees theory. Our results can be used to give unified proofs of the mating-of-trees theorems for the quantum disk and the quantum sphere, in addition to a mating-of-trees description of the weight $W = \frac{γ^2}{2}$ quantum disk. Moreover, it serves as a key ingredient in our companion work [AHS21], which proves an exact formula for $\mathrm{SLE}_κ(ρ_-;ρ_+)$ using conformal welding of random surfaces and a conformal welding result giving the so-called SLE loop.

Motivation & Objective

• To establish a finite-volume analog of the conformal welding of quantum wedges, which is fundamental to mating-of-trees theory in Liouville quantum gravity.
• To prove that welding two quantum disks of weights $W_1, W_2 > 0$ along their boundary lengths results in a quantum disk decorated with an independent $\operatorname{SLE}_{\kappa}(W_1-2;W_2-2)$ curve.
• To unify and simplify existing proofs of mating-of-trees theorems for the quantum disk and quantum sphere by leveraging the welding structure.
• To provide a foundational result for companion work on exact formulas for $\operatorname{SLE}_{\kappa}(\rho_{-};\rho_{+})$ and the SLE loop measure via conformal welding of random surfaces.

Proposed method

• The authors use a geometric construction where a quantum disk is obtained by creating and pinching a bottleneck in a quantum wedge, enabling the transfer of welding results from infinite to finite area.
• They establish that the conditional law of each resulting quantum disk given the boundary length $\ell$ is a weight-$W_i$ quantum disk conditioned on having a boundary arc of length $\ell$.
• The proof relies on coupling with imaginary geometry fields and analyzing flow lines of such fields to control the behavior of SLE curves in random surfaces.
• The authors apply total variation coupling arguments to show that the SLE curve segments are asymptotically independent from the field outside a small neighborhood, enabling the welding construction.
• They use conformal transformations and Brownian motion computations in cones and the upper half-plane to derive the scaling behavior of exit probabilities and boundary measures.
• The key technical tool is a precise computation of the boundary Poisson kernel in a wedge domain via conformal mapping to the upper half-plane, leading to an exact formula for the measure $\mu^\gamma_{\mathbb{R}_+^2}(\ell, ri)$.

Experimental results

Research questions

• RQ1 Can the conformal welding of quantum wedges be extended to the finite-area setting of quantum disks?
• RQ2 What is the law of the resulting surface when two quantum disks are welded along their boundary lengths?
• RQ3 How does the SLE curve that separates the two disks relate to the weights of the original quantum disks?
• RQ4 Can the mating-of-trees theorems for the quantum disk and sphere be derived uniformly from this welding construction?
• RQ5 What is the exact scaling behavior of the boundary measure associated with SLE curves in quantum disks?

Key findings

• Welding two quantum disks of weights $W_1, W_2 > 0$ along their boundary lengths results in a quantum disk of weight $W_1 + W_2$ decorated with an independent $\operatorname{SLE}_{\kappa}(W_1-2;W_2-2)$ curve, where $\kappa = \gamma^2$.
• The conditional law of each component quantum disk given the boundary length $\ell$ is a weight-$W_i$ quantum disk conditioned on having a boundary arc of length $\ell$.
• The joint law of the welded quantum disk and SLE curve is measurable with respect to the pair of component quantum disks, ensuring the construction is Markovian.
• The result provides a unified derivation of the mating-of-trees theorems for the quantum disk and quantum sphere, simplifying prior proofs based on pinching infinite-area surfaces.
• An exact formula is derived for the boundary measure $\mu^\gamma_{\mathbb{R}_+^2}(\ell, ri)$, showing it scales as $C \ell^{\frac{4}{\gamma^2}-1} r^{\frac{4}{\gamma^2}-1} (\ell^{\frac{4}{\gamma^2}} + r^{\frac{4}{\gamma^2}})^{-2}$.
• The method extends to the critical case $\gamma = 2$, $\kappa = 4$, via similar considerations, suggesting robustness of the welding framework.