Gaussian Multiplicative Chaos & KPZ Duality

• Gaussian multiplicative chaos is a framework that constructs random measures by exponentiating log-correlated Gaussian fields, capturing multifractal phenomena in complex systems.
• The theory delineates subcritical, critical, and supercritical regimes, with dual constructions (atomic chaos) effectively handling phase transitions when standard measures vanish.
• Its duality with the KPZ formula in Liouville quantum gravity provides a rigorous mathematical basis for understanding scaling exponents and singular random geometries.

Gaussian multiplicative chaos (GMC) is a family of random measures constructed by exponentiating log-correlated Gaussian fields, serving as foundational objects in the paper of multifractal phenomena, particularly in Liouville quantum gravity (LQG), statistical physics, random geometry, and probability. GMC theory captures the “multiplicative” scaling of geometric observables in highly irregular random environments by combining Gaussian field fluctuations and nonlinear transformations. The development of GMC encompasses canonical regular cases (“subcritical”), criticality at the phase transition, and the construction of “atomic” or “dual” chaos in regimes where the conventional theory degenerates.

1. Standard Construction and Transition Regime

The classical GMC framework begins with a log-correlated Gaussian field $X$ on $\mathbb{R}^d$, which is only defined as a generalized function with covariance kernel exhibiting logarithmic singularity. Regularizing $X$ via a sequence $(X^n)$, the standard GMC measure over a Borel set $A$ is given as the limit

$$M(A) = \lim_{n \to \infty} \int_A \exp\left( \gamma X^n(x) - \frac{\gamma^2}{2} \mathbb{E}[(X^n(x))^2] \right) dx.$$

Kahane's seminal result shows that $M$ is nontrivial if and only if $\gamma^2 < 2d$. For $\gamma^2 \geq 2d$, this measure vanishes, marking a sharp “transition” in the phase diagram (Barral et al., 2012).

In two dimensions (relevant for LQG), this threshold is at $\gamma = 2$.

2. Atomic Gaussian Multiplicative Chaos and Supercritical Regime

For $\gamma^2 > 2d$, the subcritical chaos measure degenerates to zero, necessitating an alternative construction to capture the multifractal nature typical of LQG in the “dual” or “supercritical” regime. Atomic GMC is achieved by subordinating the multiplicative chaos to an independent $\alpha$-stable random measure:

• Introduce a Poisson random measure $N_\alpha$ on $\mathbb{R}^d \times (0, \infty)$ with intensity $dx\,dz/z^{1+\alpha}$ for $\alpha \in (0,1)$. This determines a purely atomic random measure $$n_\alpha(dx) = \int_0^{+\infty} z\, N_\alpha(dx,dz)$$.
• The atomic GMC approximations are then

$$\overline{M}_n(A) = \int_A \exp\left( \frac{\gamma}{\alpha} X^n(x) - \frac{\gamma^2}{2\alpha} \mathbb{E}[(X^n(x))^2] \right) n_\alpha(dx),$$

introducing an effective dual parameter $\overline{\gamma} = \gamma/\alpha$ (Barral et al., 2012).

By coupling $\gamma$ and $\alpha$ so that $\alpha = \gamma^2/(2d)$, the “dual” chaos parameter $\overline{\gamma}$ can be taken into the regime $\overline{\gamma}^2 > 2d$—the supercritical phase.

A primary result is that $\overline{M}_n$ converges (in probability and in the sense of weak convergence of measures) to a nontrivial, purely atomic GMC measure $\overline{M}$. The Laplace transform of $\overline{M}$ is characterized via the underlying subcritical chaos $M$: $$\mathbb{E}\left[ e^{-\sum_{i=1}^p u_i \overline{M}(A_i)} \right] = \mathbb{E}\left[ e^{- \frac{\Gamma(1-\alpha)}{\alpha} \sum_{i=1}^p u_i^\alpha M(A_i)} \right].$$

3. KPZ Formula and Duality in Liouville Quantum Gravity

The Knizhnik–Polyakov–Zamolodchikov (KPZ) formula relates scaling dimensions measured with respect to Lebesgue and with respect to GMC, forming a cornerstone of the geometric interpretation in LQG: $$\dim_{\text{Leb}}(K) = \frac{\xi(\dim_M(K))}{d}, \qquad \xi(q) = \left(d + \frac{\gamma^2}{2}\right)q - \frac{\gamma^2}{2}q^2,$$ for a Borel set $K$.

For atomic (dual) chaos with parameter $\overline{\gamma}$, one obtains: $$\overline{\xi}(q) = \left( d + \frac{\overline{\gamma}^2}{2} \right)q - \frac{\overline{\gamma}^2}{2} q^2, \qquad \dim_{\text{Leb}}(K) = \frac{\overline{\xi}(\dim_{\overline{M}}(K))}{d}.$$ The structural relation $\overline{\xi}(q) = \xi(q/\alpha)$ manifests duality between the two regimes. The Hausdorff dimensions with respect to the standard and atomic measures are linearly related: $$\dim_{\overline{M}}(K) = \frac{\gamma^2}{2d} \dim_M(K) = \alpha\, \dim_M(K).$$ This duality enables the rigorous construction of LQG measures in regimes where the standard chaos is trivial, extending the mathematical definition of random geometry to the full quantum parameter space (Barral et al., 2012).

4. Singular Liouville Measures and Geometric Interpretation

The atomic chaos measure $\overline{M}$ provides a rigorous construction of so-called “singular” Liouville measures, anticipated for the dual string regime in LQG, especially for $\gamma > 2$ in two dimensions (where $\overline{\gamma} < 2$, with $\gamma\cdot\overline{\gamma}=4$).

Mathematically, $\overline{M}$ is purely atomic: its support consists of random, countable collections of points with heavy-tailed weights. This atomicity reflects the geometric singularity associated with the dual branch, such as the splitting of the quantum random geometry into “bottlenecks” or “baby universes.”

The scaling relations of the moments and the explicit Laplace functionals signal that all information in the dual chaos flows from the law of the (subcritical) standard chaos (Barral et al., 2012).

5. Summary of Main Formulas

Construct/Formula	Set-up/Result	Regime
$M(A)$	$$\lim_n \int_A e^{\gamma X^n(x)-\frac{\gamma^2}{2}\mathbb{E}[X^n(x)^2]}dx$$	$\gamma^2<2d$
$\overline{M}_n(A)$	$$\int_A e^{\frac{\gamma}{\alpha}X^n(x)-\frac{\gamma^2}{2\alpha} \mathbb{E}[X^n(x)^2]} n_\alpha(dx)$$	atomic, $\gamma^2\geq 2d$
KPZ (standard)	$\dim_{\mathrm{Leb}}(K)=\xi(\dim_M(K))/d$	$\gamma^2<2d$
KPZ (atomic/dual)	$$\dim_{\mathrm{Leb}}(K)=\overline{\xi}(\dim_{\overline{M}}(K))/d$$; $\overline{\xi}(q)=\xi(q/\alpha)$	$\gamma^2\geq 2d$
Duality of dimensions	$\dim_{\overline{M}}(K)=\alpha \dim_{M}(K)$	all regimes
Laplace functional of atomic chaos	$$\mathbb{E}[e^{-\sum u_i \overline{M}(A_i)}]=\mathbb{E}[e^{-\frac{\Gamma(1-\alpha)}{\alpha} \sum u_i^\alpha M(A_i)}]$$	atomic

6. Applications and Significance

• The atomic GMC framework rigorously extends the theory of random measures and scaling relations in random geometry to regimes inaccessible to standard chaos, particularly in the context of LQG and multifractal random measures.
• The explicit duality relations between standard and atomic chaos reinforce the connection between scaling exponents, KPZ duality, and the physical duality between “standard” and “dual” branches in conformal field theory-linked models.
• The appearance of purely atomic measures models the expected phenomena of singular coarse geometries such as bottlenecks in the geometry of quantum surfaces, giving a robust probabilistic foundation to “pinched” or disconnected random metrics (Barral et al., 2012).

7. Broader Perspectives

The construction of atomic Gaussian multiplicative chaos and the associated KPZ duality:

• Provides a unified methodology for constructing random measures in regimes where the lognormal chaos degenerate, highlighting the importance of subordination to independent stable measures.
• Deepens the mathematical underpinning of duality in quantum gravity and random geometry, with concrete consequences for the understanding of the spectrum of scaling exponents and the behavior of “singular” measures in random geometric models.
• Opens new questions regarding the fine structure of extreme singularities (“supercritical” points), interactions between random atomic measures and subcritical background, and the general mathematical landscape connecting multifractal analysis, probability, and mathematical physics (Barral et al., 2012).