Gaussian Multiplicative Chaos Measure

• Gaussian Multiplicative Chaos is a framework that defines multifractal random measures by exponentiating renormalized log-correlated Gaussian fields to overcome divergence issues.
• It employs mollification and renormalization methods, including Seneta–Heyde normalization, to ensure convergence in subcritical, critical, and supercritical regimes.
• The approach has broad applications in Liouville quantum gravity, random matrix theory, and stochastic processes, linking geometric, analytic, and probabilistic insights.

Gaussian multiplicative chaos (GMC) is the canonical construction of random measures of the form formally given by $e^{\gamma X(x)}\,dx$, where $X$ is a log-correlated Gaussian field on a domain $D \subset \mathbb{R}^d$ and $\gamma$ is a parameter. GMC provides the mathematical framework for describing multifractal random measures arising in probability, statistical physics, Liouville quantum gravity, and related fields. The construction requires care, since the oscillatory nature of $X$ and its infinite variance at small scales render the naive exponential ill-defined. Rigorous theory addresses this using regularized approximations, renormalization to control divergences, and martingale or limit techniques. The GMC allows for critical and supercritical regimes, each with distinct renormalization and limiting behavior.

1. Construction and Renormalization

The starting point is a log-correlated Gaussian field $X$ on a domain $D$, with covariance $K(x, y) = \mathbb{E}[X(x)X(y)]$ exhibiting a singularity similar to $-\log|x-y|$ plus a regular part. Since $X$ is a generalized function (distribution), one defines mollified approximations $X_\epsilon(x)$, such as convolutions with smooth kernels, or takes circle/sphere averages. The approximating GMC measure is

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

Renormalization by $-\frac{\gamma^2}{2} \mathbb{E}[X_\epsilon(x)^2]$ is critical for compensating the diverging variance. For $\gamma$ in the subcritical regime ($\gamma \in [0, \sqrt{2d})$), $M_{\gamma,\epsilon}$ converges (in probability, $L^1$ or almost surely, depending on context) to a nontrivial measure $M_\gamma$ as $\epsilon \to 0$ (Berestycki, 2015, Shamov, 2014, Aru, 2017). This convergence is robust: different regularization schemes or approximation sequences, provided their covariances are suitably close, yield the same limit in the sense of weak convergence of measures (Junnila et al., 2015). This universality is a key feature of GMC and ensures that it is determined solely by the covariance structure of $X$.

In the critical regime ($\gamma = \sqrt{2d}$), naive limits degenerate (the measure $M_{\gamma,\epsilon}$ vanishes). Instead, a renormalization of Seneta–Heyde type is applied: one introduces a diverging deterministic normalization $V_t$ and shows convergence of $V_t M_t$ to a nontrivial random measure, the derivative chaos $M'$, constructed by differentiating with respect to $\gamma$ prior to the limit (Duplantier et al., 2012). The equivalence between this derivative martingale construction and the Seneta–Heyde normalization is established rigorously, allowing one to handle the critical case robustly, including for singular base measures $\mu$ (Lacoin, 2023).

In the supercritical regime ($\gamma > \sqrt{2d}$), direct or even critically renormalized measures degenerate to zero, but after a two-parameter renormalization (involving both a power of $\epsilon$ and a logarithmic correction), stable convergence occurs to an "integrated atomic measure" whose intensity is given by the critical GMC (Bertacco et al., 10 Apr 2025).

2. Universality, Shift Invariance, and Uniqueness

A major conceptual advance is the intrinsic characterization of GMC via its shift properties. Let $X$ be a Gaussian field indexed by a Hilbert space and $Y$ a generalized function encoding the evaluation of $X$ at points of $D$. The random measure $M$ is a GMC over $(X, Y)$ if for any Cameron–Martin direction $\xi$,

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

holds almost surely (Shamov, 2014). This shift invariance axiom encapsulates the GMC's exponential response to translations of the underlying field. This framework encompasses classical constructions and also permits formulations in terms of randomized shifts: $Y$ is a randomized shift if the law of $X+Y(t)$ is absolutely continuous with respect to $X$ for $\mu$-almost every $t$. The existence of a subcritical GMC is equivalent to $Y$ being a randomized shift.

The uniqueness of the limit measure is a central theme. If two approximation sequences of log-correlated Gaussian fields have covariances close in a uniform sense, then the respective GMC measures converge to the same random measure, as long as (at least one of) the limits are non-atomic (Junnila et al., 2015). This guarantees that the GMC is intrinsic to the log-correlated structure rather than artifacts of approximation.

An elementary argument adapting these ideas shows that for any regularization, the limiting chaos measure is supported on the points where the field realizes its "thickness" (so-called thick points), and the construction yields a universal limit independent of the specifics of the mollifier or expansion (Berestycki, 2015).

3. Moment Estimates and Multifractal Structure

GMC measures exhibit a rich multifractal structure. Moment calculations—often via Kahane’s convexity inequalities—show that, in the subcritical regime, the positive moments of the measure $(M_\gamma(f))^q$ exist for $q < 2d / \gamma^2$ and negative moments exist as well (Duplantier et al., 2012, Aru, 2017). The $q$th moment scaling exponent $\zeta(q)$ is

$$\zeta(q) = (2d + \gamma^2/2)q - (d \gamma^2 / 2)q^2$$

for the 2D Gaussian free field (GFF) and more generally obtained from the covariance.

In the critical case, moments of the derivative chaos $M'$ exist for $q < 1$ and negative moments as well (Duplantier et al., 2012). These moment estimates enable precise power-law control of measures of balls, covering numbers, and energy integrals—establishing exact local dimensions and dimension conservation under projections (Falconer et al., 2016).

KPZ-type relations provide a quadratic formula connecting the Euclidean Hausdorff dimension of a set to its "quantum" dimension under the GMC measure. At criticality, for the 2D GFF, the KPZ formula reads

$\dim_{leb}(K) = 4\dim_{M'}(K) - 2 [\dim_{M'}(K)]^2$

where $\dim_{M'}(K)$ denotes the Hausdorff dimension with respect to the critical GMC measure (Duplantier et al., 2012). Analogous KPZ relations hold in other critical and subcritical contexts, relating geometry in the base space to that in the random geometry defined by GMC.

4. Applications: Liouville Quantum Gravity, Random Matrix Theory, and Stochastic Processes

GMC is fundamental in the probabilistic construction of Liouville quantum gravity (LQG), describing random two-dimensional metric geometries. For the 2D Gaussian free field in a domain $D$ with Dirichlet boundary conditions, the critical Liouville measure is constructed by integrating an extra conformal radius factor,

$M_{x,D}(dx) = C(x, D)^2 M'(dx)$

to ensure conformal covariance (Duplantier et al., 2012). This leads to conformally invariant random measures, crucial in LQG, and the same moment and KPZ relations apply.

In random matrix theory, the exponential of the log-modulus of the characteristic polynomial of ensembles such as the GUE, after suitable normalization, converges (in the L$^2$ regime) to a GMC measure (Berestycki et al., 2017, Kivimae, 2020). The approach uses asymptotic analysis of Hankel determinants with Fisher–Hartwig singularities, the Riemann–Hilbert problem, and detailed asymptotic formulas.

For stochastic processes, GMC measures serve as random clocks in time-changed Markov processes (e.g., Liouville Brownian motion). In the L$^2$-regime, convergence of the time-changed process follows from convergence of the underlying Markov process and the associated chaos measures (Ooi, 2023). Extensions encompass limits of random walks, $\alpha$-stable processes under GMC time-changes, and stable convergence results for non-Gaussian multiplicative chaos (Kim et al., 25 Oct 2024, Chowdhury et al., 24 Feb 2025).

5. Projection, Harmonic, and Dimensional Properties

Exact dimensionality results show that if the base measure $\nu$ is exact dimensional of dimension $\alpha$, and $\gamma^2/2 < \alpha$, then the corresponding GMC measure has dimension $\alpha - \gamma^2/2$ (Falconer et al., 2016). For the LQG measure $\mu$ in two dimensions (with Lebesgue base), this yields dimension $2-\gamma^2/2$. For $\gamma$ sufficiently small, projections of the GMC measure in all directions are absolutely continuous with Hölder continuous densities and positive Fourier dimension (Falconer et al., 2016).

Recent work quantifies the Fourier dimension of one-dimensional subcritical GMC, establishing (confirming the Garban–Vargas conjecture) that, almost surely, the Fourier dimension equals the multifractal (correlation) dimension, depending explicitly on $\gamma$ (Lin et al., 21 Nov 2024). On the unit circle, the GMC measure is almost surely a Rajchman measure: its Fourier coefficients vanish at infinity (Garban et al., 2023). For small $\gamma$, rescaled coefficients converge in law to a complex Gaussian law modulated by a GMC measure (Brownian modulation), connecting to complex analysis and potential theory.

In complex analysis, GMC measures serve as Clark measures for random inner functions on the disc. The induced analytic function is almost surely a Blaschke product, with precise phase transition behavior in the density of zeros (Huang et al., 2023).

6. Extensions: Non-Gaussian, Singular, and Supercritical Regimes

The general GMC framework adapts to non-Gaussian log-correlated fields by coupling constructions and universality principles. Under mild regularity and exponential moment assumptions, chaos measures for non-Gaussian and Gaussian log-correlated fields are mutually absolutely continuous throughout the entire subcritical regime (Kim et al., 25 Oct 2024, Chowdhury et al., 24 Feb 2025). Techniques rely on Skorokhod embeddings, uniform moment control, and advanced probabilistic methods.

For reference measures $\mu$ that are singular or not Lebesgue, critical GMC can still be constructed provided $\mu$ is sufficiently spread out. Precise integrability conditions (involving integrals of $|x-y|^{-d} e^{-\rho(\log \frac{1}{|x-y|})}$) ensure non-degeneracy, and examples demonstrate the near-optimality of such criteria (Lacoin, 2023).

In the supercritical regime, where the naive GMC construction is trivial, properly renormalized measures converge (stably) to integrated atomic measures with the spatial intensity given by critical GMC. The limiting measure is of the form $P_\gamma[\mu_{\gamma_c}]$—a Poissonian sum of Dirac masses with random intensities—unique up to a multiplicative constant depending only on regularization (Bertacco et al., 10 Apr 2025).

7. Analytic, Probabilistic, and Physical Significance

GMC provides the rigorous probabilistic backbone for multifractal structures in Liouville theory, random planar maps, quantum gravity, and turbulence models. Moment calculations, intermittency expansions, and connections with special functions yield exact laws (e.g., Mellin transforms, Selberg/Morris distributions) for total masses in specific settings (Ostrovsky, 2016, Remy et al., 2018).

Small deviation estimates for GMC masses underpin probabilistic path integral formulations of the massless Sinh–Gordon model and provide sharp bounds for the existence and behavior of free energies in quantum field theory (Barashkov et al., 29 Aug 2024). These probabilistic arguments replace divergent formal path integrals with well-defined GMC-based constructions and connect geometric function theory, harmonic analysis, and stochastic processes under a unified measure-theoretic framework.

GMC’s shift invariance, universality, and renormalization principles, combined with a suite of analytic and probabilistic techniques, have established it as the standard paradigm for the analysis of log-correlated random measures with applications throughout modern mathematics and physics.