dc.contributor.author Gwynne, Ewain en dc.contributor.author Miller, Jason en dc.date.accessioned 2020-09-11T23:31:33Z dc.date.available 2020-09-11T23:31:33Z dc.identifier.issn 0246-0203 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/310201 dc.description.abstract For $\gamma \in (0,2)$, $U\subset \BB C$, and an instance $h$ of the Gaussian free field (GFF) on $U$, the $\gamma$-Liouville quantum gravity (LQG) surface associated with $(U,h)$ is formally described by the Riemannian metric tensor $e^{\gamma h} (dx^2 + dy^2)$ on $U$. Previous work by the authors showed that one can define a canonical metric (distance function) $D_h$ on $U$ associated with a $\gamma$-LQG surface. We show that this metric is conformally covariant in the sense that it respects the coordinate change formula for $\gamma$-LQG surfaces. That is, if $U,\widetilde{U}$ are domains, $\phi \colon U \to \widetilde{U}$ is a conformal transformation, $Q=2/\gamma+\gamma/2$, and $\widetilde h = h\circ\phi^{-1} + Q\log|(\phi^{-1})'|$, then $D_h(z,w) = D_{\widetilde{h}}(\phi(z),\phi(w))$ for all $z,w \in U$. This proves that $D_h$ is intrinsic to the quantum surface structure of $(U,h)$, i.e., it does not depend on the particular choice of parameterization. dc.publisher Institute of Mathematical Statistics dc.rights All rights reserved dc.rights.uri dc.title Conformal covariance of the Liouville quantum gravity metric en dc.type Article prism.publicationName L'Institut Henri Poincare, Annales B: Probabilites et Statistiques en dc.identifier.doi 10.17863/CAM.57287 dcterms.dateAccepted 2020-09-09 en rioxxterms.version AM rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved en rioxxterms.licenseref.startdate 2020-09-09 en rioxxterms.type Journal Article/Review en pubs.funder-project-id European Commission Horizon 2020 (H2020) ERC (804166) cam.orpheus.counter 21 * rioxxterms.freetoread.startdate 2023-09-11
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