Liouville quantum gravity and KPZ. (English) Zbl 1226.81241

Duplantier, Bertrand; Sheffield, Scott

Invent. Math. 185, No. 2, 333-393 (2011).

Authors’ abstract: Consider a bounded planar domain \(D\), an instance \(h\) of the Gaussian free field on \(D\), with Dirichlet energy \((2\pi )^{-1}\int _{D }\nabla h(z)\cdot \nabla h(z)dz\), and a constant \(0\leq \gamma <2\). The Liouville quantum gravity measure on \(D\) is the weak limit as \(\varepsilon \rightarrow 0\) of the measures
\[ \varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz, \]
where \(dz\) is the Lebesgue measure on \(D\) and \(h _{\varepsilon }(z)\) denotes the mean value of \(h\) on the circle of radius \(\varepsilon\) centered at \(z\). Given a random (or deterministic) subset \(X\) of \(D\) one can define the scaling dimension of \(X\) using either the Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov [“Fractal structure of 2D-quantum gravity”, Mod. Phys. Lett. A 3, 819–826 (1988)] relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of \(\partial D\)). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

Reviewer: Gil de Oliveira-Neto (Minas Gerais)

MSC:

81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
28A33	Spaces of measures, convergence of measures
52C20	Tilings in \(2\) dimensions (aspects of discrete geometry)
60A10	Probabilistic measure theory
60K40	Other physical applications of random processes
83C80	Analogues of general relativity in lower dimensions

Keywords:

Liouville quantum gravity; probability measures; KPZ relation

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