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• W2942932816 abstract "We show that for each $gamma in (0,2)$, there is a unique metric (i.e., distance function) associated with $gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) $h$, there is a unique random metric $D_h$ associated with the Riemannian metric tensor $e^{gamma h} (dx^2 + dy^2)$ on $mathbb C$ which is characterized by a certain list of axioms: it is locally determined by $h$ and it transforms appropriately when either adding a continuous function to $h$ or applying a conformal automorphism of $mathbb C$ (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The $gamma$-LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding, Dubedat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when $gamma = sqrt{8/3}$, our metric coincides with the $sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general $gamma in (0,2)$, we conjecture that our metric is the Gromov-Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems." @default.
• W2942932816 created "2019-05-09" @default.
• W2942932816 creator A5019995974 @default.
• W2942932816 creator A5035621341 @default.
• W2942932816 date "2019-05-01" @default.
• W2942932816 modified "2023-09-27" @default.
• W2942932816 title "Existence and uniqueness of the Liouville quantum gravity metric for $gamma in (0,2)$" @default.
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• W2942932816 hasPublicationYear "2019" @default.
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