Conformal welding for critical Liouville quantum gravity

Holden, Nina; Powell, Ellen

Abstract

Consider two critical Liouville quantum gravity surfaces (i.e., γ-LQG for γ = 2), each with the topology of H and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces, when the boundaries are identified according to quantum boundary length. This results in a critical LQG surface decorated by an independent SLE4. Combined with the proof of uniqueness for such a welding, recently established by McEnteggart, Miller, and Qian (2018), this shows that the welding operation is well-defined. Our result is a critical analogue of Sheffield’s quantum gravity zipper theorem (2016), which shows that a similar conformal welding for subcritical LQG (i.e., γ-LQG for γ ∈ (0, 2)) is well-defined.

Citation

Holden, N., & Powell, E. (2021). Conformal welding for critical Liouville quantum gravity. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 57(3), 1229-1254. https://doi.org/10.1214/20-aihp1116

• Journal Article Type: Article
• Acceptance Date: Oct 22, 2020
• Online Publication Date: Jul 22, 2021
• Publication Date: 2021-08
• Deposit Date: Dec 9, 2020
• Publicly Available Date: Aug 5, 2021
• Journal: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
• Print ISSN: 0246-0203
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 57
• Issue: 3
• Pages: 1229-1254
• DOI: https://doi.org/10.1214/20-aihp1116
• Related Public URLs: https://arxiv.org/abs/1812.11808