Gaussian Multiplicative Chaos Measure

• Gaussian multiplicative chaos is a universal probabilistic measure constructed by exponentiating and renormalizing a log‐correlated Gaussian field.
• It exhibits distinct regimes: subcritical yielding non-degenerate atomless measures, critical via derivative martingale or Seneta–Heyde renormalization, and supercritical leading to purely atomic measures.
• GMC plays a crucial role in Liouville quantum gravity, random matrix theory, and multifractal analysis, linking probabilistic behavior with geometric and physical phenomena.

A Gaussian multiplicative chaos (GMC) random measure is a universal probabilistic object constructed by exponentiating a regularized log-correlated Gaussian field, appropriately renormalized. GMC describes the random limit measures arising from such exponential interactions and exhibits a rich phase diagram depending on the "coupling" parameter. Its study unites probability theory, mathematical physics, geometric measure theory, and several connections to Liouville quantum gravity and random matrix theory.

1. Log-Correlated Gaussian Fields and Regularization

The foundation of GMC is the log-correlated Gaussian field $X$ on a domain $D\subset\mathbb{R}^d$, formally characterized by covariance

$\E\left[X(x) X(y)\right] = \log \frac{1}{|x-y|} + g(x,y)$

where $g$ is continuous on $D \times D$. Such a field is only defined as a random Schwartz distribution (generalized function), so regularization is essential. This is performed by mollification: for a mollifier $\theta$,

$$X_\varepsilon(x) = \int_D X(y)\, \theta_\varepsilon(x-y) \,\mathrm{d}y,\quad \theta_\varepsilon(x) = \varepsilon^{-d}\theta(x/\varepsilon)$$

which smooths $X$ at scale $\varepsilon$. The variance has the essential logarithmic divergence $\Var[X_\varepsilon(x)] = \log(1/\varepsilon) + O(1)$ uniformly in $x$ (Berestycki, 2015).

2. Definition, Phase Transition, and Construction of GMC

For $\gamma > 0$ and reference measure $\mu$ (Lebesgue or more general Radon measures), define the random measures: $M_\varepsilon^{\gamma}(\mathrm{d}x) = \exp\big( \gamma X_\varepsilon(x) - \tfrac{\gamma^2}{2} \E[X_\varepsilon(x)^2] \big)\, \mu(\mathrm{d}x)$ A critical phenomenon occurs in the parameter $\gamma$:

• If $\gamma^2 < 2d$, this family of measures converges in probability (and in $L^1$) as $\varepsilon \to 0$ to a non-degenerate, atomless random Radon measure $M^{\gamma}$.
• If $\gamma^2 \geq 2d$, the limit becomes trivial: $M^{\gamma} \equiv 0$ almost surely (Berestycki, 2015, Shamov, 2014).

Uniform integrability is established via second-moment computations: $\E[M_\varepsilon^\gamma(D)^2] = \iint_{D\times D} e^{\gamma^2\Cov(X_\varepsilon(x),X_\varepsilon(y))} \mathrm{d}x \mathrm{d}y < \infty \Longleftrightarrow \gamma^2 < 2d$ (Berestycki, 2015).

The universality (independence from mollifier) is rigorously established, e.g., via Karhunen-Loève expansion or by shift-covariant (randomized shift) properties (Berestycki, 2015, Shamov, 2014).

3. Critical and Supercritical Gaussian Multiplicative Chaos

At the threshold $\gamma_c = \sqrt{2d}$ (criticality), naive regularization yields a vanishing measure. Two alternative constructions yield a non-trivial limit (Duplantier et al., 2012, Duplantier et al., 2012):

• Derivative martingale: Take the derivative in $\gamma$ at criticality:

$D_\varepsilon(\mathrm{d}x) = [ - X_\varepsilon(x) + \gamma_c \E[X_\varepsilon(x)^2] ] \exp\left( \gamma_c X_\varepsilon(x) - \frac{\gamma_c^2}{2} \E[X_\varepsilon(x)^2] \right) \mathrm{d}x$

These converge a.s. to a positive, non-atomic, atomless measure $M'$, with full support and universal scaling properties (Duplantier et al., 2012).

• Seneta-Heyde renormalization: Multiply the vanishing measure by $\sqrt{\log (1/\varepsilon)}$, i.e.,

$\widetilde{M}_\varepsilon^{\gamma_c}(\mathrm{d} x) = \sqrt{\log(1/\varepsilon)}\, \exp\left( \gamma_c X_\varepsilon(x) - \frac{\gamma_c^2}{2} \E[X_\varepsilon(x)^2] \right) \mathrm{d}x$

Both constructions yield the same critical chaos measure up to explicit constants (Duplantier et al., 2012).

In the supercritical regime $\gamma^2 > 2d$, the only nontrivial limiting random objects are purely atomic random measures, constructed as Poisson clusterings with intensity the critical chaos. The limit is universal up to a multiplicative constant and independent of regularization (Bertacco et al., 10 Apr 2025). The limiting atomic measure is characterized as follows: $M^\gamma = \sum_{k \ge 1} Z_k \delta_{x_k}$ where $(x_k)$ are points from a Poisson process on $D$ with intensity $M^{\gamma_c}$ and $Z_k$ are independent random masses with Pareto-type tails.

4. Uniqueness, Universality, and Shift Equivariance

Shamov's general framework rigorously establishes the uniqueness and universality of subcritical GMC (Shamov, 2014). A random measure $M$ on $T$ is a (subcritical) GMC over a Gaussian field $(X, Y)$ with expectation $\mu$ if:

• $M$ is measurable w.r.t.\ $X$
• $M \ll \mu$ a.s., $\E M = \mu$
• Shift-covariance (randomized shift): for any Cameron–Martin vector $\xi \in H$,

$$M(X+\xi,\,dt) = \exp(\langle Y(t),\xi\rangle) M(X,dt)$$

For log-correlated fields, this shift-covariance captures the structural essence of GMC; universality follows since any two regularization schemes generate equivalent (in law, and even almost surely as functions of the underlying field) random measures (Shamov, 2014).

5. Dimensional, Geometric, and Regularity Properties

The limiting GMC measure $M^\gamma$ is non-atomic and multifractal. Important quantitative properties include:

• Moments: For $p \in \left(0, 2d/\gamma^2\right)$,

$\E[M^\gamma(A)^p] < \infty$

• Exact-dimensionality: For a base measure $\nu$ of dimension $\alpha$, the GMC measure $\widetilde{\nu}$ is almost surely exact-dimensional of dimension $\alpha - \gamma^2/2$ provided $\gamma^2/2 < \alpha$ (Falconer et al., 2016).
• Projections: For sufficiently small $\gamma$ (e.g., $\gamma<0.28$ in dimension $2$), all orthogonal projections of Liouville quantum gravity (LQG) measure are absolutely continuous with Hölder continuous densities (Falconer et al., 2016).
• Fourier dimension: The GMC measure, due to Hölder-regularity of projections, has positive Fourier (Salem) dimension almost surely (Falconer et al., 2016).
• Complex GMC: Analytic continuation to complex $\gamma$ yields a random generalized function valued in Sobolev/Besov spaces, with sharp moment and regularity bounds in admissible sectors of the complex plane (Lacoin, 2023, Junnila et al., 2019).

6. Geometric and Physical Applications; Liouville Quantum Gravity

GMC is the foundation of Liouville quantum gravity (LQG), describing random geometry in 2D quantum gravity (Berestycki, 2015, Duplantier et al., 2012). The random measure constructed by exponentiating the Gaussian free field (GFF) is the area measure in LQG. The critical and supercritical regimes correspond to the boundary and glassy phases in LQG, with the KPZ formula relating Euclidean and quantum dimensions: $$\dim_{\mathrm{Leb}}(K) = (1 + \gamma^2/4) \dim_{M^\gamma}(K) - (\gamma^2/4) (\dim_{M^\gamma}(K))^2$$ and for the critical measure $M'$, the map is $\Phi(q)=d q (2-q)$ (Duplantier et al., 2012).

In the supercritical/atomic phase, the KPZ duality and Poisson cluster structure provide a rigorous model for the glassy, freezing phase of 2D random geometry (Barral et al., 2012, Bertacco et al., 10 Apr 2025).

7. Connections and Universality Beyond Gaussianity

Recent developments rigorously establish universality of GMC in the subcritical regime for a large class of non-Gaussian, log-correlated fields, notably via invariance principles and coupling techniques (Kim et al., 25 Oct 2024, Chowdhury et al., 24 Feb 2025). For i.i.d. exponentials, or random characteristic polynomials of random matrices (unitary, orthogonal, symplectic), convergence to GMC has been proved in the appropriate regime (Berestycki et al., 2017, Forkel et al., 2020). This universality indicates that GMC is the canonical scaling limit of multiplicative cascades and a universal object for extremes of log-correlated fields.

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Summary Table: GMC Regimes and Limiting Measures

$\gamma^2$	Limiting GMC Measure	Structure	Construction
$0 < \gamma^2 < 2d$	$M^\gamma$	Atomless, universal	Regularization + renormalization
$\gamma^2 = 2d$	$M'$ (derivative chaos)	Atomless, universal	Derivative martingale/Seneta–Heyde
$\gamma^2 > 2d$	$M^\gamma$ (atomic)	Purely atomic, PPP	Poisson cluster over $M^{\gamma_c}$

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• An Elementary Approach to Gaussian Multiplicative Chaos (Berestycki, 2015)
• On Gaussian Multiplicative Chaos (Shamov, 2014)
• Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale (Duplantier et al., 2012)
• Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Formula (Duplantier et al., 2012)
• Complex Gaussian Multiplicative Chaos on Phase Boundaries (Lacoin, 2023)
• Exact Dimensionality and Projection Properties of Gaussian Multiplicative Chaos (Falconer et al., 2016)
• Uniqueness of Supercritical Gaussian Multiplicative Chaos (Bertacco et al., 10 Apr 2025)
• Gaussian Multiplicative Chaos and KPZ Duality (Barral et al., 2012)
• Invariance Principle for the Gaussian Multiplicative Chaos via a High Dimensional CLT with Low Rank Increments (Chowdhury et al., 24 Feb 2025)
• Absolute Continuity of Non-Gaussian and Gaussian Multiplicative Chaos Measures (Kim et al., 25 Oct 2024)