Subsequential Scaling Limits for Liouville Graph Distance

Abstract

For $0<\gamma<2$ and $\delta>0$, we consider the Liouville graph distance, which is the minimal number of Euclidean balls of $\gamma$-Liouville quantum gravity measure at most $\delta$ whose union contains a continuous path between two endpoints. In this paper, we show that the renormalized distance is tight and thus has subsequential scaling limits at $\delta\to 0$. In particular, we show that for all $\delta>0$ the diameter with respect to the Liouville graph distance has the same order as the typical distance between two endpoints.