{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/362540"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/362540","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Random conformally covariant metrics in the plane","abstract":"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.","abstract_html":"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 &quot;<span class=\"etd-inline-math\">e<sup>&gamma; h</sup> (dx<sup>2</sup>+dy<sup>2</sup>)</span>&#x27;&#x27; where $h$ is a Gaussian free field (GFF) on a planar domain and <span class=\"etd-inline-math\">&gamma; \\in (0,2)</span>. Duplantier and Sheffield constructed the <span class=\"etd-inline-math\">&gamma;</span>-LQG area and boundary length measures, which fall under the framework of Kahane&#x27;s Gaussian multiplicative chaos. Later, a conformally covariant distance metric associated to <span class=\"etd-inline-math\">&gamma;</span>-LQG was constructed for whole-plane and zero-boundary GFFs. In this thesis we describe the <span class=\"etd-inline-math\">&gamma;</span>-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 <span class=\"etd-inline-math\">&gamma; = \\sqrt{8/3}</span> (for which the LQG metric can be explicitly described in terms of Brownian motion) to the entire subcritical range <span class=\"etd-inline-math\">&gamma; \\in (0,2)</span>. 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.","abstract_has_math":true,"creators":["Hughes, Liam"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Miller, Jason"],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-09-28","date_published":"2023-09-28","updated_at":"2026-07-22T22:24:21Z","subjects":["probability"],"languages":["eng"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/296977e5-d681-40b4-b7fc-8f57c03c915d/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.104670","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Miller, Jason"]},{"key":"dc:creator","label":"Author","values":["Hughes, Liam"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2023-09-28"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/362540"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["probability"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/296977e5-d681-40b4-b7fc-8f57c03c915d/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.17863/CAM.104670"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/1101d907-0a90-458b-8205-2749b0c9b825/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["87eda9de84448d1f82354d60eee3eb5f","4f8cf0e0a14f80ea7edf14fbef514df4"]},{"key":"dc:title","label":"Title","values":["Random conformally covariant metrics in the plane"]}]}],"canonical_facts":{"dc:contributor.advisor":["Miller, Jason"],"dc:creator":["Hughes, Liam"],"dc:date.issued":["2023-09-28"],"dc:description.abstract":["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."],"dc:format.checksum.md5":["87eda9de84448d1f82354d60eee3eb5f","4f8cf0e0a14f80ea7edf14fbef514df4"],"dc:identifier.doi":["https://doi.org/10.17863/CAM.104670"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/1101d907-0a90-458b-8205-2749b0c9b825/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/362540"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/296977e5-d681-40b4-b7fc-8f57c03c915d/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["probability"],"dc:title":["Random conformally covariant metrics in the plane"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:21Z"}