Bounds for distances and geodesic dimension in Liouville first passage percolation

Abstract

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 $\xi \geq 0$. For $\xi \leq 2/d_{2}$, this leads to new bounds for $d_{\gamma }$ which improve on the best previously known upper (resp. lower) bounds for $d_{\gamma }$ in the case when $\gamma > \sqrt{8/3} $ (resp. $\gamma \in (0.4981, \sqrt{8/3} )$). These bounds are consistent with the Watabiki (1993) prediction for $d_{\gamma }$. However, for $\xi > 1/\sqrt{3} $ (or equivalently for LQG with central charge larger than 17) our bounds are inconsistent with the analytic continuation of Watabiki’s prediction to the $\xi >2/d_{2}$ regime. We also obtain an upper bound for the Euclidean dimension of LFPP geodesics.