Regularity and confluence of geodesics for the supercritical Liouville quantum gravity metric

Let $h$ be the planar Gaussian free field and let $D_h$ be a supercritical Liouville quantum gravity (LQG) metric associated with $h$. Such metrics arise as subsequential scaling limits of supercritical Liouville first passage percolation (Ding-Gwynne, 2020) and correspond to values of the matter central charge $\mathbf{c}_{\mathrm M} \in (1,25)$. We show that a.s. the boundary of each complementary connected component of a $D_h$-metric ball is a Jordan curve and is compact and finite-dimensional with respect to $D_h$. This is in contrast to the \emph{whole} boundary of the $D_h$-metric ball, which is non-compact and infinite-dimensional with respect to $D_h$ (Pfeffer, 2021). Using our regularity results for boundaries of complementary connected components of $D_h$-metric balls, we extend the confluence of geodesics results of Gwynne-Miller (2019) to the case of supercritical Liouville quantum gravity. These results show that two $D_h$-geodesics with the same starting point and different target points coincide for a non-trivial initial time interval.