AI 中文总结
该研究联系刘维尔引力与随机暴胀，通过物理推导得到KPZ标度关系，揭示刘维尔体积重正化崩溃对应永恒暴胀转变，还探讨了与随机曲面数学概率方法的关联。
AI 中文摘要
受暴胀和刘维尔理论的启发，我们研究由整体标度因子 $ds^2 = e^{2 \zeta} dx^2$ 表征的随机 $d$ 维几何，该几何由具有对数关联的随机场 $\zeta(x)$ 给出。我们讨论体积元 $e^{d \zeta}$ 的重正化相关问题，将著名的刘维尔理论公式（KPZ）与暴胀理论公式联系起来。通过从固定的物理截断出发，将算子粗粒化至由平坦度规 $dx^2$ 定义的固定基准截断，我们直接从物理角度推导了KPZ标度关系。我们指出，随机暴胀中会出现相同的重正化问题，该问题由暴胀子场的布朗运动建模。我们证明，当涨落幅度超过临界值时，刘维尔体积重正化的崩溃对应着向永恒暴胀的转变；在 $d=2$ 时，这与刘维尔引力中熟知的 $c_m = 1$ 阈值相匹配。我们还讨论了其与随机曲面的数学概率方法的联系。
英文摘要
Motivated by inflation and Liouville theory, we consider random $d$-dimensional geometries characterized by an overall scale factor $ds^2 = e^{2 ζ} dx^2 $ given in terms of a random Gaussian field $ζ(x)$ with logarithmic correlations. We discuss aspects of the renormalization of the volume element $e^{d ζ}$, connecting well-known Liouville theory formulas (KPZ) and inflationary ones. By starting from a fixed physical cutoff and coarse-graining operators to a fixed fiducial cutoff, defined via the flat metric $dx^2$, we provide a direct physical derivation of the KPZ scaling relation. We point out that the same renormalization problem arises in stochastic inflation, where it is modeled by Brownian motion of the inflaton field. We show that the breakdown of the renormalization of the Liouville volume when the fluctuation amplitude exceeds a critical value corresponds to the transition to eternal inflation. In $d = 2$, this matches the familiar $c_m = 1$ barrier in Liouville gravity. We also discuss connections to mathematical probabilistic approaches to random surfaces.
Comments34 pages plus appendices, 8 figures
