Abstract

We define a random metric associated to Liouville quantum gravity (LQG) for all values of matter central charge $c<25$ by extending the axioms for a weak LQG metric from the $c<1$ setting. We show that the axioms are satisfied by subsequential limits of Liouville first passage percolation; Ding and Gwynne (2020) showed these limits exist in a suitably chosen topology. We show that, in contrast to the $c<1$ phase, the metrics for $c\in (1,25)$ do not induce the Euclidean topology since they a.s. have a dense (measure zero) set of singular points, points at infinite distance from all other points. We use this fact to prove that a.s. the metric ball is not compact and its boundary has infinite Hausdorff dimension. On the other hand, we extend many fundamental properties of LQG metrics for $c<1$ to all $c\in (-\infty ,25)$, such as a version of the (geometric) Knizhnik–Polyakov–Zamolodchikov (KPZ) formula.

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