on Liouville quantum gravity and KPZ

Overview

Duplantier and Sheffield gave a rigorous construction of Liouville quantum gravity area measure and proved the KPZ relation between Euclidean and quantum scaling exponents (Duplantier & Sheffield, 2010). Their work made precise a central prediction from two-dimensional quantum gravity and conformal field theory.

The proof combines Gaussian free fields, multiplicative chaos, Brownian circle averages, and Frostman-type dimension estimates.

Random Geometry in Two Dimensions

A smooth conformal metric on a planar domain can be written as

$$d{s}^2=e^{\gamma h(z)}|dz{|}^2,$$

where $h$ is a scalar field and $\gamma$ is a parameter. In Liouville quantum gravity, $h$ is not a smooth function. It is a Gaussian free field, a random distribution.

The formal area element is

$$e^{\gamma h(z)}dz.$$

This expression is meaningless pointwise because $h(z)$ is not defined. The first task is to define it by regularization and renormalization.

Let $h_{\epsilon }(z)$ be a circle average of the Gaussian free field around $z$ at radius $\epsilon$. Its variance grows like

$$\mathrm{Var}(h_{\epsilon }(z))\sim \log {\epsilon }^{-1}.$$

Therefore $e^{\gamma h_{\epsilon }(z)}$ diverges in expectation. The correct renormalized measure is

$${\mu }_{\gamma ,\epsilon }(dz)={\epsilon }^{{\gamma }^2/2}{e}^{\gamma h_{\epsilon }(z)}dz.$$

For $0\le \gamma <2$, these measures converge in probability and in suitable senses to a nontrivial random measure

$${\mu }_{\gamma }.$$

Gaussian multiplicative chaos

The construction is an instance of Gaussian multiplicative chaos. The renormalizing factor subtracts the variance explosion, since

$$\mathbb{E}{e}^{\gamma h_{\epsilon }(z)}=e^{{\gamma }^2\mathrm{Var}(h_{\epsilon }(z))/2}.$$

Euclidean and Quantum Scaling

Let $X$ be a deterministic fractal set independent of the field $h$. Its Euclidean scaling exponent $x$ is related to its Euclidean dimension by

$${\dim }_E(X)=2-2x.$$

The quantum scaling exponent $\Delta$ measures how the quantum area needed to hit $X$ scales. Informally, if a quantum ball has quantum area $\delta$, then the probability that it intersects $X$ behaves like

$${\delta }^{\Delta }.$$

The KPZ formula predicts

$$x=\frac{{\gamma }^2}{4}{\Delta }^2+\left(1-\frac{{\gamma }^2}{4}\right)\Delta .$$

This is a nonlinear change of dimension caused by the random geometry. When $\gamma =0$, quantum area is just Euclidean area and the formula reduces to $x=\Delta$.

KPZ relation

For a deterministic Borel set $X$ independent of the Gaussian free field, the Euclidean scaling exponent $x$ and Liouville quantum gravity scaling exponent $\Delta$ satisfy

$$x=\frac{{\gamma }^2}{4}{\Delta }^2+\left(1-\frac{{\gamma }^2}{4}\right)\Delta$$

for $0\le \gamma <2$.

Circle Averages as Brownian Motion

Fix a point $z$. As the radius changes logarithmically,

$$t=\log {\epsilon }^{-1},$$

the circle average process behaves like Brownian motion in $t$:

$$h_{e^{-t}}(z)\approx B_t.$$

Thus the quantum mass of a Euclidean ball of radius $e^{-t}$ is roughly

$${\mu }_{\gamma }(B_{e^{-t}}(z))\approx \exp \left(\gamma B_t-\frac{{\gamma }^2}{2}t\right)e^{-2t},$$

up to multiplicative fluctuations controlled by the field away from the circle-average part.

This converts geometric scaling into a problem about Brownian motion with drift. The KPZ quadratic comes from solving the relation between Euclidean radius and quantum mass when the log-mass contains the Brownian term $\gamma B_t$.

Moment Estimates

A central estimate controls moments of the quantum mass of small Euclidean balls:

$$\mathbb{E}\left[{\mu }_{\gamma }(B_r(z){)}^q\right]\asymp r^{\zeta (q)},$$

where

$$\zeta (q)=\left(2+\frac{{\gamma }^2}{2}\right)q-\frac{{\gamma }^2}{2}{q}^2$$

for the relevant range of $q$. The concavity of $\zeta$ reflects intermittency: rare thick points of the Gaussian free field contribute disproportionately to high moments.

These moment estimates provide the bridge between coverings in Euclidean geometry and coverings measured by quantum mass.

Proof: KPZ via Covering Exponents

To prove one inequality, cover $X$ by Euclidean balls $B_{r_i}(z_i)$ and estimate powers of their quantum masses:

$$\mathbb{E}{\mu }_{\gamma }(B_{r_i}(z_i){)}^q\asymp r_i^{\zeta (q)}.$$

If the Euclidean covering exponent of $X$ is $x$, then sums of the form ${\sum }_i{r}_i^{2x+ϵ}$ can be made small. Choosing $q$ so that $\zeta (q)=2x+ϵ$ converts this Euclidean covering into a quantum covering whose $q$-mass is summable. This gives the upper bound for the quantum exponent after optimizing over $q$.

For the reverse inequality, take a Frostman measure $\nu$ supported on $X$ with finite Euclidean energy at every exponent below the critical one. Weighting $\nu$ by the circle-average field produces a random measure supported on $X$. Its expected quantum energy can be written as a double integral in $z,w$, where the logarithmic covariance of the Gaussian free field contributes a power of $|z-w|$. The same exponent $\zeta$ appears, now as the condition under which the expected energy is finite. Positive probability of finite energy gives the matching lower bound for the quantum dimension.

The Brownian circle-average calculation explains the algebra behind this optimization. Under the rooted measure biased by the Liouville mass near a point, the circle-average increment becomes Brownian motion with drift. If

$$A=-\frac{1}{\gamma }\log \delta ,a=\frac{2}{\gamma }-\frac{\gamma }{2},$$

the Euclidean time corresponding to quantum mass $\delta$ is modeled by the hitting time

$$T_A=\inf \{t:B_t+at=A\}.$$

A Euclidean set of exponent $x$ is hit at scale $e^{-T_A}$ with probability roughly $\exp (-2x{T}_A)$. The exponential martingale for Brownian motion gives

$$\mathbb{E}{e}^{-2x{T}_A}=e^{-\beta A}$$

precisely when

$$2x=\beta a+\frac{{\beta }^2}{2}.$$

Setting $\Delta =\beta /\gamma$ gives

$$x=\frac{{\gamma }^2}{4}{\Delta }^2+\left(1-\frac{{\gamma }^2}{4}\right)\Delta .$$

The covering and Frostman arguments make this hitting-time computation rigorous and identify the exact exponent transformation.

Conceptual Meaning

Liouville quantum gravity changes the ruler. A Euclidean ball of radius $r$ has random quantum mass governed by both its area $r^2$ and the field average over the ball. Points where the field is unusually high make small Euclidean regions quantum-large. Fractal sets are therefore reweighted according to how often they intersect thick points of the field.

The KPZ formula is universal in the sense that it depends only on $\gamma$ and the Euclidean exponent, not on detailed geometry of $X$, as long as $X$ is deterministic or independent of the field.

Metric versus measure

The Duplantier-Sheffield paper constructs and studies the random area measure and exponent relation. A full metric-space theory of Liouville quantum gravity requires additional work beyond this measure-level construction.

Gaussian Free Field Preliminaries

The Gaussian free field on a planar domain is a centered Gaussian random distribution whose covariance is the Green function. It is not defined pointwise, but its circle average

$$h_{\epsilon }(z)$$

is a genuine Gaussian random variable. The logarithmic singularity of the Green function gives

$$\mathrm{Cov}(h_{\epsilon }(z),h_{\delta }(z))\approx \log \frac{1}{\max (\epsilon ,\delta )}.$$

This is why the circle-average process in logarithmic time behaves like Brownian motion.

A point is called $\alpha$-thick if

$$h_{\epsilon }(z)\sim \alpha \log {\epsilon }^{-1}$$

as $\epsilon \downarrow 0$. Thick points are rare, but they dominate the high moments of the Liouville measure. The KPZ relation is ultimately a dimension conversion formula that accounts for how often a deterministic set intersects these thick points.

From Quantum Balls to Exponents

A Euclidean ball of radius $r=e^{-t}$ has quantum mass of order

$$\exp \left(\gamma B_t-\frac{{\gamma }^2}{2}t\right)e^{-2t}.$$

Solving for the Euclidean scale that corresponds to quantum mass $\delta$ introduces a Brownian hitting-time problem with drift. The quadratic term in KPZ is the variance contribution of this Brownian fluctuation; the linear term is the deterministic area scaling corrected by renormalization.

The proof avoids defining a full random metric ball. Instead, it uses quantum mass of Euclidean balls and covering exponents. This is why the theorem is a measure-level KPZ theorem: it rigorously identifies how fractal dimensions transform under the random area measure, without requiring a completed metric theory of LQG.

Heuristic Derivation of the Quadratic

Suppose a Euclidean ball of radius $e^{-t}$ has quantum mass approximately

$$\exp \left(\gamma B_t-\frac{{\gamma }^2}{2}t-2t\right).$$

A set with Euclidean exponent $x$ is hit by such a ball with probability about $e^{-2xt}$. If a quantum ball has mass $\delta =e^{-q}$, then the relevant Euclidean time $t$ is random and is determined by

$$q\approx \left(2+\frac{{\gamma }^2}{2}\right)t-\gamma B_t.$$

The probability of hitting the set is then governed by an exponential moment of this Brownian hitting time.

Solving the associated exponential martingale calculation produces

$$x=\frac{{\gamma }^2}{4}{\Delta }^2+\left(1-\frac{{\gamma }^2}{4}\right)\Delta .$$

The rigorous proof replaces this heuristic by covering and Frostman estimates, but the Brownian hitting-time computation explains why the answer is quadratic rather than linear.

Transferable Mechanisms

The Liouville measure construction belongs to the larger theory of Gaussian multiplicative chaos, surveyed by Rhodes and Vargas (Rhodes & Vargas, 2014). That framework validates the renormalized exponential

$${\epsilon }^{{\gamma }^2/2}{e}^{\gamma h_{\epsilon }(z)}dz$$

as part of a general mechanism for building random measures from log-correlated fields.

For other problems, the portable idea is to separate a rough field into scale increments and then control the resulting random measure through moment exponents. The same perspective appears in thick points, random planar maps, branching random walks, and extrema of log-correlated fields, where geometry is governed by rare but statistically quantifiable high points.

Links

• on Ising models --- Liouville quantum gravity is part of the continuum probability framework surrounding two-dimensional critical lattice models.
• on cover times for two-dimensional random walks --- both topics use logarithmically correlated fields and thick-point phenomena in two dimensions.
• on random polynomial zeros --- Gaussian analytic objects and random measures appear in both settings, though the geometric questions are different.

References

🐻  Duplantier, B. & Sheffield, S. 2010. Liouville Quantum Gravity and KPZ. Inventiones Mathematicae 185(2), 333–393.

🐻  Rhodes, R. & Vargas, V. 2014. Gaussian Multiplicative Chaos and Applications: A Review. Probability Surveys 11, 315–392.