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

中文翻译：

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星尺度不变场的刘维尔度量：尾部和外尔尺度

我们研究了与具有短程相关性的对数相关高斯场的近似值相关的 Liouville 度量。我们表明，在参数 $$\gamma _c >0$$ γ c > 0 下，黎曼度量的矩形的左右长度 $$e^{\gamma \phi _{0,n}} ds^2 $$ e γ ϕ 0 , nds 2 各种纵横比集中于准对数正态尾，当 $$\gamma < \min ( \gamma _c, 0.4)$$ γ < min ( γ c , 0.4 ) 并且后续限制与 Weyl 标度一致。