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 ${\hat{I}}_c^3>0$, the left-right length of rectangles for the Riemannian metric $e^{{\hat{I}}^3\ddot{I}{•}_{0,n}}d{s}^2$ with various aspect ratio is concentrated with quasi-lognormal tails, that the renormalized metric is tight when ${\hat{I}}^3<\min ({\hat{I}}_c^3,0.4)$ and that subsequential limits are consistent with the Weyl scaling.

Final version. To appear in PTRF. 52 pages, 8 figures