Gaussian Multiplicative Chaos

• Gaussian Multiplicative Chaos is a probabilistic framework defining random measures by exponentiating log‐correlated Gaussian fields, crucial for modeling turbulence and Liouville quantum gravity.
• It exhibits multifractal scaling laws with precise moment estimates, offering insights into critical regimes and harmonic analysis through renormalized limits.
• GMC applies to random matrix theory, network models, and quantum gravity, while ongoing research investigates its Fourier decay properties and universality across diverse settings.

Gaussian Multiplicative Chaos (GMC) is a probabilistic framework rigorously defining random measures generated by exponentiating log-correlated Gaussian fields. Originally motivated by models of turbulence and Liouville quantum gravity, GMC provides a universal object capturing multifractal structure and extreme value statistics in a range of settings, including planar statistics, random matrix theory, and critical polymer models. The essential feature is the renormalized limit

$$M_\gamma(dx) = \lim_{\varepsilon \to 0} \exp\left( \gamma X_\varepsilon(x) - \frac{\gamma^2}{2} \mathbb E[X_\varepsilon(x)^2] \right) \sigma(dx)$$

where $X_\varepsilon$ is a mollification of a centered Gaussian field $X$ with covariance kernel logarithmically singular on the diagonal, and $\gamma>0$ is the so-called intermittency or temperature parameter.

1. Definition, Construction, and Uniqueness

GMC is constructed from a log-correlated Gaussian field $X$ defined, for example, by

$\mathbb E[X(x) X(y)] = -\log|x-y| + O(1)$

on a suitable domain or measure space $(T, \sigma)$. The field is only a distribution, so approximation (such as convolution or circle averages) is necessary. The regularized chaos (for $\gamma$ in the subcritical regime) is defined by

$$M_{\gamma,\varepsilon}(dx) = \exp\left( \gamma X_\varepsilon(x) - \frac{\gamma^2}{2} \mathbb E[X_\varepsilon(x)^2]\right) \sigma(dx)$$

which, as $\varepsilon \to 0$, converges (in probability or almost surely) to a nontrivial random measure $X_\varepsilon$0 for $X_\varepsilon$1 less than a critical value $X_\varepsilon$2 (for $X_\varepsilon$3-models with Lebesgue $X_\varepsilon$4, $X_\varepsilon$5) (Aru, 2017, Shamov, 2014, Junnila et al., 2015, Chowdhury et al., 24 Feb 2025).

Pathwise uniqueness holds: the GMC measure is uniquely determined as a measurable function of the field, independent of the approximation scheme, provided the approximating covariances converge appropriately (bounded overall and vanishing off-diagonal difference) and the limit is non-atomic (Junnila et al., 2015, Shamov, 2014). The construction extends to general metric measure spaces and to cases with singular base measures, fractal supports, or conditional random base measures (Lacoin, 2023, Garban et al., 2018, Falconer et al., 2016).

2. Multifractal Properties and Moments

The GMC measure exhibits multifractal scaling:

$X_\varepsilon$6

with structure function

$X_\varepsilon$7

and Hausdorff dimension of the support $X_\varepsilon$8 for Lebesgue base measure. For a general base measure of dimension $X_\varepsilon$9, the induced GMC measure is almost surely exact-dimensional with dimension $X$0, as long as $X$1 (Aru, 2017, Falconer et al., 2016, Ghosh et al., 2020).

All negative moments of the total GMC mass exist below criticality, uniformly in the regularity of the base measure, provided its $X$2-energy is finite ($X$3, with $X$4) (Garban et al., 2018, Aru, 2017). Positive moments exist up to a critical value $X$5.

On the unit interval, exact formulas for fractional moments of total mass, including cases with insertion points (logarithmic singularities) at the endpoints, are given via the Selberg integral and its continuation in the subcritical regime $X$6 (Remy et al., 2018). For the circle, the total mass and moments are also described explicitly and connected to physics-derived conjectures (Chhaibi et al., 2019).

3. Harmonic and Geometric Analysis

A central topic concerns Fourier analysis of GMC, especially on the circle or planar domains. For the circle-GMC, almost surely the measure is Rajchman: all Fourier coefficients decay to zero at infinity, with explicit moment-based decay rates for $X$7 and a precise law for the renormalized coefficients (Garban et al., 2023). The Fourier dimension is bounded above by $X$8, but for Hausdorff dimension $X$9; whether the measure is Salem (i.e., the two dimensions coincide) remains open (Garban et al., 2023, Falconer et al., 2016).

Projection properties have been established: for sufficiently small $\gamma>0$0, all orthogonal projections of planar GMC are absolutely continuous and possess Hölder continuous densities, with the measure possessing positive Fourier dimension almost surely (Falconer et al., 2016). In higher dimensions and infinite-dimensional Wiener spaces, GMC exhibits geometric effects such as exponential decay of mass in balls and robust localization in low-temperature or strong disorder regimes (Ghosh et al., 2020, Bröker et al., 2018).

4. Critical and Complex GMC

In the critical regime $\gamma>0$1, the naive construction degenerates and requires Seneta–Heyde or derivative-martingale renormalization. Three definitions—renormalized mollification, critical martingale, and derivative martingale—all converge to the same critical GMC under a sharp spatial integrability criterion on the base measure (Lacoin, 2023). The critical case has universal 1/t tail decay for the total mass (Wong, 2019), with explicit law for the circle mass as the reciprocal of an exponential (Chhaibi et al., 2019).

Complex GMC, where the exponent $\gamma>0$2 is not real, leads to a rich phase diagram. At specific boundaries between "phases," limit measures exist as complex Gaussian white noises with random intensities dictated by real critical chaos, or, in other regimes, as $\gamma>0$3-limits. At the so-called "freezing transition," the limit becomes a purely atomic measure—mirroring glassy phases in physical systems (Lacoin, 2023).

5. Universality, Coupling, and Extensions

GMC possesses strong universality properties. Log-correlated non-Gaussian chaos (random Fourier series with i.i.d. non-Gaussian coefficients) yields random measures almost surely mutually absolutely continuous with the canonical Gaussian GMC in the entire subcritical regime, as established via explicit couplings relying on high-dimensional CLT and matched support at "thick points" (Chowdhury et al., 24 Feb 2025, Kim et al., 2024). This extends to random wave or random matrix-like models, and also underlies the equivalence with certain circular beta ensembles (Chhaibi et al., 2019).

Conditional GMC constructions are essential in settings where standard chaos fails, such as continuum random polymers on critical fractals or (2+1)-dimensional polymer models. There, the limit measure at a later time is a conditional GMC over the (random) measure at an earlier time, with the corresponding Gaussian field sampled independently but with intersection local time entering the covariance structure (Clark, 2019).

6. Applications in Mathematical Physics and Network Models

GMC measures are fundamental in 2D Liouville quantum gravity, giving the canonical volume measure. The KPZ relation algebraically links Euclidean and GMC scaling dimensions and characterizes multifractality in random geometry (Aru, 2017). In random matrix theory, GMC appears as the limit law of characteristic polynomials or log-modulus fields, with explicit connections for moments and total mass (Remy et al., 2018, Chhaibi et al., 2019).

In stochastic networks, the latent spatial structure of GMC drives the formation of Fractal Gaussian Networks (FGNs), flexible sparse random geometric graphs with analyzable scaling laws for motif and clique counts. Parameterization by the fractality parameter $\gamma>0$4 interpolates from random geometric graphs to strongly inhomogeneous, fractal structures observed empirically in real-world networks (Ghosh et al., 2020).

7. Open Problems and Advanced Directions

Several outstanding questions persist:

• The Fourier dimension of GMC and fine Salem properties remain unresolved for most regimes (Garban et al., 2023, Falconer et al., 2016).
• The behavior at phase boundaries and in the "frozen" region of complex GMC is only partially characterized (Lacoin, 2023).
• The microscopic structure of critical and supercritical measures in higher dimensions, and their universality classes, require further analysis (Lacoin, 2023, Wong, 2019).
• Conditional and multi-layer GMC structures are conjectured to capture critical continuum directed polymers in two dimensions (Clark, 2019).
• Dynamical and geometric properties in infinite-dimensional settings (Wiener chaos, stochastic heat equations) continue to be an active topic (Bröker et al., 2020, Bröker et al., 2018).

The theory of Gaussian multiplicative chaos thus provides a unified probabilistic mechanism at the intersection of analysis, probability, and mathematical physics, supporting both deep universal properties and fine fractal analysis across a spectrum of models.