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.

65 pages, 10 figures