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

Abstract

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.