Bounds for distances and geodesic dimension in Liouville first passage percolation

Understand

For $\xi \geq 0$, Liouville first passage percolation (LFPP) is the random metric on $\varepsilon \mathbb Z^2$ obtained by weighting each vertex by $\varepsilon e^{\xi h_\varepsilon(z)}$, where $h_\varepsilon(z)$ is the average of the whole-plane Gaussian free field $h$ over the circle $\partial B_\varepsilon(z)$.

• Ding and Gwynne (2018) showed that for $\gamma \in (0,2)$, LFPP with parameter $\xi = \gamma/d_\gamma$ is related to $\gamma$-Liouville quantum gravity (LQG), where $d_\gamma$ is the $\gamma$-LQG dimension exponent.
• For $\xi > 2/d_2$, LFPP is instead expected to be related to LQG with central charge greater than 1.
• We prove several estimates for LFPP distances for general $\xi\geq 0$.