Volume of metric balls in Liouville quantum gravity

Understand

We study the volume of metric balls in Liouville quantum gravity (LQG).

• For $\gamma \in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for $p \in (-\infty, 4/\gamma^2)$.
• Here, we prove that the LQG volume of LQG metric balls admits all finite moments.
• This answers a question of Gwynne and Miller and generalizes a result obtained by Le Gall for the Brownian map, namely, the $\gamma = \sqrt{8/3}$ case.