  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.
