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Existence and uniqueness of the Liouville quantum gravity metric for γ ∈ (0, 2)

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We show that for each $\gamma \in (0,2)$, there is a unique metric 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 $D_h = ``e (dx2 + dy^2)”$ 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 $\BB C$ (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 \emph{Liouville first passage percolation} (LFPP), the random metric obtained by exponentiating a mollified version of the GFF . Earlier work by Ding, Dub\’edat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. We show 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.

Based on four joint papers with Jason Miller and one joint paper with Julien Dub\’edat, Hugo Falconet, Josh Pfeffer, and Xin Sun.

This talk is part of the Probability series.

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