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.
Citation
Ewain Gwynne. Joshua Pfeffer. "Bounds for distances and geodesic dimension in Liouville first passage percolation." Electron. Commun. Probab. 24 1 - 12, 2019. https://doi.org/10.1214/19-ECP248
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