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Random conformally covariant metrics in the plane https://doi.org/10.17863/CAM.104670 Hughes, Liam Miller, Jason 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 "$e^{\gamma h} (dx^2+dy^2)$'' where $h$ is a Gaussian free field (GFF) on a planar domain and $\gamma \in (0,2)$. Duplantier and Sheffield constructed the $\gamma$-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 $\gamma$-LQG was constructed for whole-plane and zero-boundary GFFs. In this thesis we describe the $\gamma$-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 $\gamma = \sqrt{8/3}$ (for which the LQG metric can be explicitly described in terms of Brownian motion) to the entire subcritical range $\gamma \in (0,2)$. 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. University of Cambridge 2023-09-28 Thesis Doctoral Doctor of Philosophy (PhD) eng https://www.repository.cam.ac.uk/handle/1810/362540 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/296977e5-d681-40b4-b7fc-8f57c03c915d/download 87eda9de84448d1f82354d60eee3eb5f https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/1101d907-0a90-458b-8205-2749b0c9b825/download 4f8cf0e0a14f80ea7edf14fbef514df4 https://www.rioxx.net/licenses/all-rights-reserved/ probability