[Paper Review] Integrability of SLE via conformal welding of random surfaces

Morris Ang, Nina Holden+1 more|arXiv (Cornell University)|Apr 19, 2021

Stochastic processes and statistical mechanics67 references5 citations

TL;DR

This paper establishes an exact formula for the moment generating function of the conformal derivative ψ′(1) in SLEκ(ρ−; ρ+), a classical variant of Schramm-Loewner evolution, by combining Liouville conformal field theory (LCFT), mating-of-trees encoding, and conformal welding of random quantum surfaces. The key result expresses E[ψ′(1)λ] in terms of the double gamma function F(α, κ, ρ−, ρ+), providing a new integrability result for SLE via LCFT and quantum geometry.

ABSTRACT

We demonstrate how to obtain integrable results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called $\mathrm{SLE}_κ(ρ_-;ρ_+)$. Our proof is built on two connections between SLE, LCFT, and mating-of-trees. Firstly, LCFT and mating-of-trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call the uniform embedding of an LQG surface, we extend earlier equivalence results by allowing fewer marked points and more generic singularities. Secondly, the conformal welding of these random surfaces produces SLE curves as their interfaces. In particular, we rely on the conformal welding results proved in our companion paper [AHS20]. Our paper is an essential part of a program proving integrability results for SLE, LCFT, and mating-of-trees based on these two connections.

Motivation & Objective

• To establish new integrability results for Schramm-Loewner evolution (SLE) by connecting it to Liouville conformal field theory (LCFT) and mating-of-trees frameworks.
• To extend the equivalence between LCFT and mating-of-trees descriptions of random surfaces in Liouville quantum gravity (LQG) to surfaces with fewer marked points and more general singularities.
• To use conformal welding of finite-volume quantum surfaces to realize SLE curves as interfaces, enabling exact computations.
• To derive an exact formula for the law of the conformal derivative ψ′(1) in SLEκ(ρ−; ρ+) using the moment generating function.
• To unify and generalize previous results on quantum surfaces and SLE via the uniform embedding of LQG surfaces and conformal welding techniques.

Proposed method

• Utilizes the uniform embedding of quantum surfaces to provide a unified, conceptual framework for describing random LQG surfaces via LCFT and mating-of-trees.
• Applies conformal welding of finite-volume quantum surfaces—established in the companion paper [AHS20]—to generate SLE curves as interfaces between glued surfaces.
• Employs the mating-of-trees encoding to describe SLE on LQG backgrounds via Brownian motion and related stochastic processes.
• Leverages integrability results from Remy and Zhu [RZ20a, RZ20b] on boundary LCFT to compute expectations involving quantum measures.
• Derives the moment generating function of ψ′(1) by combining conformal welding with LCFT path integral formalism and quantum surface invariance under coordinate changes.
• Uses the coordinate change formula for γ-LQG: f •γ h = h ◦ f⁻¹ + Q log |(f⁻¹)′|, to relate different embeddings of quantum surfaces and maintain invariance under conformal maps.

Experimental results

Research questions

• RQ1 How can integrability results for SLE be derived from Liouville conformal field theory and mating-of-trees?
• RQ2 What is the exact distribution of the conformal derivative ψ′(1) for SLEκ(ρ−; ρ+) in terms of its moment generating function?
• RQ3 How do LCFT and mating-of-trees provide equivalent yet complementary descriptions of random quantum surfaces in LQG?
• RQ4 In what way does conformal welding of quantum surfaces produce SLE curves as their interfaces?
• RQ5 What is the role of the double gamma function F(α, κ, ρ−, ρ+) in encoding the law of ψ′(1) for SLEκ(ρ−; ρ+)?

Key findings

• The paper derives an exact formula for the moment generating function of the conformal derivative ψ′(1) in SLEκ(ρ−; ρ+), showing that E[ψ′(1)λ] = F(α, κ, ρ−, ρ+) / F(√κ, κ, ρ−, ρ+) for λ < λ₀, where λ₀ = (1/κ)(ρ₊ + 2)(ρ₊ + 4 − κ/2).
• For λ ≥ λ₀, the moment E[ψ′(1)λ] diverges, indicating a critical threshold in the tail behavior of the conformal derivative.
• The function F(α, κ, ρ−, ρ+) is defined using the double gamma function Γb(z), with b = √κ/2, and encodes the exact law of ψ′(1) in terms of the SLE parameters.
• The result is independent of the choice of solution α to the quadratic equation 1 − (α/2)(√κ/2 + 2/√κ − α/2) = λ, ensuring consistency of the moment generating function.
• The derivation relies on the conformal welding of quantum surfaces from [AHS20], which realizes SLE curves as interfaces between glued LQG surfaces.
• The proof unifies LCFT and mating-of-trees perspectives by introducing the uniform embedding of quantum surfaces, extending earlier results to surfaces with fewer marked points and more general singularities.