Liouville metric of star-scale invariant fields: tails and Weyl scaling

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 $e^{\gamma {\varphi }_{0,n}}d{s}^2$ 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.