Random conformally covariant metrics in the plane

This thesis is in the broad area of random conformal geometry, combining tools from probability and complex analysis. We mainly consider *Liouville quantum gravity* (LQG), a model introduced in the physics literature in the 1980s by Polyakov in order to provide a canonical example of a random surface with conformal symmetries and formally given by the Riemannian metric tensor "e^{γ h} (dx²+dy²)'' where $h$ is a Gaussian free field (GFF) on a planar domain and γ \in (0,2). Duplantier and Sheffield constructed the γ-LQG area and boundary length measures, which fall under the framework of Kahane's Gaussian multiplicative chaos. Later, a conformally covariant distance metric associated to γ-LQG was constructed for whole-plane and zero-boundary GFFs. In this thesis we describe the γ-LQG metric corresponding to a free-boundary GFF and derive basic properties and estimates for the boundary behaviour of the metric using GFF techniques. We use these to show that when one uses a conformal welding to glue together boundary segments of two appropriate independent LQG surfaces to get another LQG surface decorated by a *Schramm--Loewner evolution* (SLE) curve, the LQG metric on the resulting surface can be obtained as a natural metric space quotient of those on the two original surfaces. This generalizes results of Gwynne and Miller in the special case γ = \sqrt{8/3} (for which the LQG metric can be explicitly described in terms of Brownian motion) to the entire subcritical range γ \in (0,2). Moreover, we show that LQG metrics are infinite-dimensional (in the sense of Assouad) and thus that their embeddings into the plane cannot be quasisymmetric. We also consider chemical distance metrics associated to *conformal loop ensembles*, the loop version of SLE, using the imaginary geometry coupling to the GFF to bound the exponent governing the conformal symmetries of such a metric.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2023

Author dc:creator

• Hughes, Liam

Advisor dc:contributor.advisor

• Miller, Jason

• probability

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/296977e5-d681-40b4-b7fc-8f57c03c915d/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.104670

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/362540