  Given a planar continuum Gaussian free field $h^{\mathcal U}$ in a domain
$\mathcal U$ with Dirichlet boundary condition and any $\delta>0$, we let
$\{h_\delta^{\mathcal U}(v): v\in \mathcal U\}$ be a real-valued smooth
Gaussian process where $h_\delta^{\mathcal U}(v)$ is the average of
$h^{\mathcal U}$ along a circle of radius $\delta$ with center $v$. For $\gamma
> 0$, we study the Liouville first passage percolation (in scale $\delta$),
i.e., the shortest path distance in $\mathcal U$ where the weight of each path
$P$ is given by $\int_P \mathrm{e}^{\gamma h_\delta^{\mathcal U}(z)} |dz|$. We
show that the distance between two typical points is $O(\delta^{c^*
\gamma^{4/3}/\log \gamma^{-1}})$ for all sufficiently small but fixed
$\gamma>0$ and some constant $c^* > 0$. In addition, we obtain similar upper
bounds on the Liouville first passage percolation for discrete Gaussian free
fields, as well as the Liouville graph distance which roughly speaking is the
minimal number of Euclidean balls with comparable Liouville quantum gravity
measure whose union contains a continuous path between two endpoints. Our
results contradict with some reasonable interpretations of Watabiki's
prediction (1993) on the random distance of Liouville quantum gravity at high
temperatures.
