KPZ formula derived from Liouville heat kernel
نویسندگان
چکیده
In this paper, we establish the Knizhnik–Polyakov–Zamolodchikov (KPZ) formula of Liouville quantum gravity, using the heat kernel of Liouville Brownian motion. This derivation of the KPZ formula was first suggested by F. David and M. Bauer in order to get a geometrically more intrinsic way of measuring the dimension of sets in Liouville quantum gravity. We also provide a careful study of the (no)-doubling behaviour of the Liouville measures in the appendix, which is of independent interest.
منابع مشابه
Gaussian multiplicative chaos and KPZ duality
This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ2 beyond the transition phase (i.e. γ2 > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a sim...
متن کاملLiouville Quantum Gravity on the Riemann sphere
In this paper, we rigorously construct 2d Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov Quantum Geometry of bosonic strings. We also establish some of its fundamental properties like conformal covariance under PSL2(C)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula for Liouville Quant...
متن کاملKPZ in one dimensional random geometry of multiplicative cascades
We prove a formula relating the Hausdorff dimension of a subset of the unit interval and the Hausdorff dimension of the same set with respect to a random path matric on the interval, which is generated using a multiplicative cascade. When the random variables generating the cascade are exponentials of Gaussians, the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum gra...
متن کاملIntroduction to Kpz
1. A physical introduction 2 1.1. KPZ/Stochastic Burgers/Scaling exponent 2 1.2. Physical derivation 3 1.3. Scaling 3 1.4. Formal invariance of Brownian motion 4 1.5. Dynamic scaling exponent 6 1.6. Renormalization of the nonlinear term 6 1.7. Cutoff KPZ models 7 1.8. Hopf-Cole solutions 8 1.9. Directed polymers in a random environment 11 1.10. Fluctuation breakthroughs of 1999 12 1.11. The Air...
متن کاملSharp Gradient Estimate and Yau’s Liouville Theorem for the Heat Equation on Noncompact Manifolds
We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton’s estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau’s celebrated Liouville theorem for positive harmonic functions...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. London Math. Society
دوره 94 شماره
صفحات -
تاریخ انتشار 2016