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

arXiv commentFinal version. To appear in PTRF. 52 pages, 8 figures

References in corpus (6)

• Uniqueness and universality of the Brownian map
• The topological structure of scaling limits of large planar maps
• Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric
• The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds
• Lognormal scale invariant random measures
• Heat kernel for Liouville Brownian motion and Liouville graph distance

Cited by in corpus (2)

• Heat kernel for Liouville Brownian motion and Liouville graph distance
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