We define a random metric associated to Liouville quantum gravity (LQG) for all values of matter
central charge
by extending the axioms for a weak LQG metric from the
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
phase, the
metrics for
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
to
all
,
such as a version of the (geometric) Knizhnik–Polyakov–Zamolodchikov (KPZ)
formula.
