Liouville Quantum Gravity Measure

• Liouville Quantum Gravity measure is a probabilistic framework using Gaussian multiplicative chaos to rigorously define random area measures on 2D surfaces.
• It exhibits conformal covariance and multifractality, distinguishing subcritical, critical, and supercritical regimes with clear implications for scaling limits.
• Its construction bridges connections with random matrix theory, SLE, and discrete planar map models, offering insights into fractal geometry in quantum gravity.

Liouville Quantum Gravity (LQG) measure refers to a family of intrinsically random measures on two-dimensional manifolds defined via exponential functionals of the Gaussian free field (GFF), appropriately regularized. These measures encode the volume element of a probabilistically defined random geometry that arises as the scaling limit of random planar maps and as the geometric framework for Liouville conformal field theory. Subcritical, critical, and supercritical regimes are distinguished according to the parameter controlling the strength of the exponential. The theory of the LQG measure encompasses its rigorous construction via Gaussian multiplicative chaos (GMC), properties under conformal transformations, moment estimates, connections to random matrix theory and SLE, and extensions to fractals and higher genus.

1. Definition and Regularization

Let $D \subset \mathbb{C}$ be a planar domain and $h$ a (generalized function-valued) Gaussian free field on $D$. The Liouville quantum gravity area measure, or LQG measure, is formally written as

$\mu_h(dz) = e^{\gamma h(z)}\,dz$

where $\gamma \in (0,2)$ is the LQG coupling constant. Since $h$ is only defined as a distribution, this expression requires regularization and renormalization. For $z \in D$ and $\varepsilon > 0$ small, define the circle-average $h_\varepsilon(z)$ as the mean of $h$ on $\partial B_\varepsilon(z)$. The regularized measure is

$$\mu_h^{(\varepsilon)}(dz) := \varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)} dz$$

The multiplicative renormalization $\varepsilon^{\gamma^2/2}$ exactly cancels the anticipated leading divergence $$E[e^{\gamma h_\varepsilon(z)}] \asymp \varepsilon^{-\gamma^2/2}$$. The limit

$$\mu_h(dz) := \lim_{\varepsilon \to 0} \mu_h^{(\varepsilon)}(dz)$$

exists almost surely along a vanishing sequence in the topology of measures and defines a non-degenerate random measure precisely for $\gamma \in (0,2)$ (Berestycki et al., 2014). This is the canonical construction of two-dimensional Gaussian multiplicative chaos (GMC).

The critical case $\gamma=2$ is singular: the normalized fields tend to zero. In this regime, the derivative normalization (Seneta–Heyde norming) yields a nontrivial critical measure $$M'(dx) = \lim_{\varepsilon \to 0} [2 \ln(1/\varepsilon) - X_\varepsilon(x) + o(1)] \varepsilon^2 e^{2 X_\varepsilon(x)} dx$$ (Rhodes et al., 2013).

For more general backgrounds, e.g., closed surfaces, the GFF is constructed via spectral theory, and the Liouville field $\phi = X + H$ (GFF plus deterministic background solution) enters the action

$$S_L[\phi] = \frac{1}{4\pi} \int_M \bigg\{ |\nabla_{g_0} \phi|^2 + Q R_{g_0} \phi + 4\pi \mu e^{\beta \phi} \bigg\} dV_{g_0}$$

with the LQG measure defined via a suitably renormalized exponential of $\phi$ (Oh et al., 2020).

2. Structure, Covariance, and Equivalence

The LQG measure is almost surely supported on a random fractal set of non-integer Hausdorff dimension, strictly less than the Euclidean dimension. For the canonical LQG measure on the domain, subcriticality ($\gamma < 2$) ensures that it is finite on compacts; for $\gamma \to 2$ one must renormalize, and in the supercritical regime no locally finite, field-local, coordinate-covariant measure exists (Bhatia et al., 16 Oct 2024).

Conformal covariance is a central property. If $\phi : \tilde{D} \to D$ is a conformal map and $Q = 2/\gamma + \gamma/2$,

$\tilde{h} = h \circ \phi + Q \log |\phi'|$

then

$\mu_{\tilde{h}}(A) = \mu_{h}(\phi(A))$

almost surely. Remarkably, the field-measure correspondence is almost surely compatible with all conformal embeddings simultaneously (Sheffield et al., 2016).

The GFF $h$ almost surely determines $\mu_h$, and vice versa: $\mu_h$ almost surely determines $h$ (Berestycki et al., 2014). Explicitly, for small $\varepsilon$, the relation

$$H_\varepsilon(z) := \gamma^{-1}\log \mu_h(B_\varepsilon(z)) = h_\varepsilon(z) + O(1)$$

shows that field averages can be reconstructed from the measure. For fractal subsets (e.g., SLE traces or Brownian paths), the associated GMC measure $\mu_{\sigma,h}$ determines the harmonic extension of $h$ off the fractal under mild regularity conditions.

3. Moments, Multifractality, and Dimensional Properties

For $d$-dimensional Frostman measures $\sigma$ (finite $\int \int |x-y|^{-d} \sigma(dx)\sigma(dy)<\infty$), the chaos measure $\mu_{\sigma,h}$ for $\gamma < \sqrt{2d}$ admits all positive moments up to $2d/\gamma^2$ and negative moments of all order (Berestycki et al., 2014). No scale-invariance assumption is needed, in contrast to classical Kahane’s theory.

The pointwise dimension of the LQG measure, for $\gamma<2$, is $2-\gamma^2/2$ almost surely on typical points. For critical GMC ($\gamma=2$), the measure is non-atomic, supported on a dense set of Hausdorff dimension zero, and does not charge polar sets (Rhodes et al., 2013). The multifractal spectrum is determined by quadratic functions parameterizing the scaling exponents.

4. Extensions: Fractals, Higher Genus, Discrete Models, and Stochastic Quantization

The construction generalizes to measures supported on deterministic or random fractal subsets (e.g., occupation measure of Brownian motion, SLE traces), resulting in chaos measures $\mu_{\sigma,h}$ tailored to the set’s dimension $d$ (Berestycki et al., 2014).

On compact surfaces of arbitrary genus, the GFF is constructed spectrally, and the Liouville GMC measure arises via mollification and subtraction of the local variance. Stochastic quantization of Liouville CFT corresponds to a nonlinear stochastic heat equation with exponential nonlinearity, whose invariant measure is the LQG measure (Oh et al., 2020).

Discrete-to-continuous transitions are rigorously described. For example, for discretized tori $\mathbb{T}_L^n$, the associated discrete polyharmonic GFF and LQG measure converge in distribution to the continuum construction as $L \to \infty$, for $|\gamma| < \sqrt{2n}$ (Schiavo et al., 2023).

5. Metric Geometry and Minkowski Content

LQG also gives rise to a natural random metric, via first-passage percolation approximations or via a system of axiomatically characterized metrics $D_h$. A fundamental result establishes that the Minkowski content measured with respect to the LQG metric coincides with the LQG measure itself: for any subset $A$, the (appropriately scaled) covering number of balls of small LQG metric-radius converges to $\mu_h(A)$. As a consequence, the metric structure $D_h$ a.s. determines the conformal structure (i.e., the quantum surface $(h, \mu_h)$) for $\gamma \in (0,2)$ (Gwynne et al., 2022).

6. Physical Regimes and Non-Existence in the Supercritical Phase

Three distinct regimes are recognized:

• Subcritical ($\gamma < 2$): The LQG measure is nontrivial, coordinate-covariant, and conforms to the exponential-of-GFF GMC paradigm.
• Critical ($\gamma = 2$): Standard normalization fails; a “derivative martingale” yields a singular, yet nontrivial, critical measure with zero Hausdorff dimension support (Rhodes et al., 2013).
• Supercritical ($\gamma>2$, $Q\in(0,2)$, central charge $c_L \in (1,25)$): There is no locally finite, field-local, coordinate-covariant area measure. Any such attempt yields either a null measure or infinite mass on every open set. Conditionally finite-volume supercritical LQG surfaces (e.g., finite random planar maps) converge to branched polymer (CRT) structures (Bhatia et al., 16 Oct 2024).

This demarcation underlies the geometric and probabilistic categorization of quantum gravity phases.

7. Connections to Random Matrices, SLE, and Planar Map Scaling Limits

Large $N$ limits of random matrix statistics (e.g., characteristic polynomial measures for unitary Brownian motion) converge to LQG measures, establishing a direct connection between random matrix theory and Liouville geometry (Bourgade et al., 2022). These probabilistic limits share the core “log-correlated field” structure, and their GMC limits match the analytic properties of the continuum LQG measure.

The LQG measure and associated structures also govern the scaling limits of random planar maps (e.g., quadrangulations), annular topologies (Remy, 2017), and SLE/CLE weldings (Ang et al., 25 Sep 2024). For example, unit-area LQG spheres admit equivalent constructions via Bessel process excursions, correlated Brownian loops, or (in the $\gamma = \sqrt{8/3}$ case) $3/2$-stable Lévy excursions producing a mating-of-trees description (Miller et al., 2015).

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This comprehensive framework positions the Liouville quantum gravity measure as the universal measure-theoretic foundation for random two-dimensional geometries, encoding the subtle fractal structure, conformal covariance, and multifractality that are the hallmark of canonical models in probabilistic conformal geometry.