Subsequential scaling limits for Liouville graph distance

Abstract

For $0<γ<2$ and $δ>0$, we consider the Liouville graph distance, which is the minimal number of Euclidean balls of $γ$-Liouville quantum gravity measure at most $δ$ 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 $δ\to 0$. In particular, we show that for all $δ>0$ the diameter with respect to the Liouville graph distance has the same order as the typical distance between two endpoints.