Bounds for distances and geodesic dimension in Liouville first passage percolation

Ewain Gwynne, Josh Pfeffer

Published 2019-03-22Version 1

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 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 (which is expected to be the Hausdorff dimension of the conjectural LQG metric). 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.