  We study the Liouville metric associated to an approximation of a
log-correlated Gaussian field with short range correlation. We show that below
a parameter $\gamma_c >0$, the left-right length of rectangles for the
Riemannian metric $e^{\gamma \phi_{0,n}} ds^2$ with various aspect ratio is
concentrated with quasi-lognormal tails, that the renormalized metric is tight
when $\gamma < \min ( \gamma_c, 0.4)$ and that subsequential limits are
consistent with the Weyl scaling.
