Fourier Dimensions for GMC

• The paper presents an explicit computation of the Fourier dimension for GMC measures, confirming its equivalence with the multifractal correlation dimension.
• It employs advanced martingale methods and multi-resolution analysis to derive precise local and global Fourier decay estimates.
• The approach resolves long-standing conjectures in high-dimensional settings, with significant implications for turbulence, quantum gravity, and fractal geometry.

Gaussian multiplicative chaos (GMC) is a mathematical framework for constructing random multifractal measures by exponentiating log-correlated Gaussian fields, with major applications in mathematical physics, probability, and fractal geometry. The Fourier dimension of such measures quantifies the decay rate of their Fourier transform, reflecting deep connections with their multifractal structure and harmonic regularity. The development of precise renormalization schemes, martingale methods, and multifractal analysis has enabled the explicit computation and characterization of Fourier dimensions for GMC and related chaotic measures, culminating in recent rigorous results that resolve long-standing conjectures for measures supported on high-dimensional spaces.

1. Construction of Gaussian Multiplicative Chaos and Log-Correlated Fields

The fundamental object is a log-correlated Gaussian field $X$ on a domain $D \subseteq \mathbb{R}^d$ or on the torus $\mathbb{T}^d$, characterized by the covariance

$\E[X(x)X(y)] = -\log|x-y| + f(x,y)$

where $f$ is smooth and the log-singularity dominates the small-scale behavior. The classical chaos measure $M_\gamma$ for parameter $\gamma$ is defined (after regularization) by

$M_\gamma(dt) = \lim_{\epsilon \to 0} e^{\gamma X_\epsilon(t) - (\gamma^2/2)\E[X_\epsilon(t)^2]} dt$

where $X_\epsilon$ denotes an $\epsilon$-regularization (such as circle-averages or Fourier cutoffs). The exponential “blow-up” is controlled by renormalization, leading to a singular but locally finite random measure that possesses multifractal properties (Lacoin et al., 2013).

The proper construction hinges on martingale convergence methods, $T$-martingales (in the sense of Kahane [1985, 1987]), or vector-valued martingale frameworks adapted for higher-dimensions and arbitrary covariance kernels (Shamov, 2014, Lin et al., 6 May 2025, Chen et al., 31 Jul 2025).

2. Fourier Dimension: Definition and Interpretation

The Fourier dimension $\dim_F \mu$ of a finite measure $\mu$ on $\R^d$ is defined by decay asymptotics: $$\dim_F\mu = \sup \{ s \in (0, d) : |\widehat{\mu}(\xi)| = O(|\xi|^{-s/2}) \text{ as } |\xi| \to \infty \}$$ where $$\widehat{\mu}(\xi) = \int e^{-2\pi i \xi \cdot x} \, d\mu(x)$$. The Fourier dimension reflects both the fractal geometry of the measure's support and its distribution across frequencies, capturing “harmonic regularity.” For GMC measures, the precise rate of polynomial decay typically aligns with their correlation dimension (multifractal scaling exponent) (Chen et al., 20 Sep 2024, Lin et al., 6 May 2025, Chen et al., 31 Jul 2025).

Recent advances extend this to the torus $\mathbb{T}^d$ by studying Fourier coefficients

$$\mu(\mathbf{k}) = \int_{\mathbb{T}^d} e^{-2\pi i \mathbf{k} \cdot x} \, d\mu(x)$$

and measuring their decay; the Fourier dimension is equivalently characterized in terms of the polynomial rate at which $|\mu(\mathbf{k})|$ decays as $|\mathbf{k}| \to \infty$.

3. Exact Computation and Main Results in High Dimensions

Recent work (Chen et al., 31 Jul 2025) has resolved the problem of determining the exact Fourier dimension for GMC measures on $\mathbb{T}^d$ for all $d \ge 1$, confirming that

$\dim_F(M_{\gamma, \mathbb{T}^d}) = D_{\gamma, d}$

where $D_{\gamma, d}$ is the correlation (or multifractal) dimension of the GMC measure, explicitly known from multifractal analysis (e.g., $D_{\gamma, d} = d - \gamma^2/2$ for $0 < \gamma < \sqrt{2d}$ [Bertacco 2023]). The main innovation is a decomposition of the log-correlated field into smooth, highly regular processes and the application of multi-resolution analysis to obtain local Fourier decay estimates, which are integrated globally via Pisier’s martingale type inequality for vector-valued martingales.

A summary of the formula: | Parameter Regime | Fourier Dimension $\dim_F$ | |-----------------------------------|------------------------------------| | $0 < \gamma < \sqrt{2d}$ | $d - \gamma^2/2$ | | $\sqrt{2d} < \gamma < \tilde{V}$ | $(V_{2d} - \gamma)/2$ (model dep.) |

This resolves the “dimensional barrier” in earlier works (Lin et al., 6 May 2025) for $d \ge 3$ and proves that the Fourier decay exponent coincides with the multifractal correlation dimension for all standard values of the GMC parameter $\gamma$ in the full sub-critical range.

4. Methodological Advances: Multi-Resolution Analysis and Vector-Valued Martingales

The proof employs a novel multi-resolution decomposition of the log-correlated field as a sum of independent smooth processes with high regularity—achieved via mollification by convolution kernels with good decay estimates [Brezis 2011]. Fourier decay is established first locally on dyadic cubes (where overlapping higher frequencies see significant cancellation due to the smoothness), and then the global bound is achieved by combining these local estimates using Pisier’s martingale type inequality in Banach spaces [Pisier 2016], which allows precise control over the aggregation of fluctuations at all scales (Chen et al., 31 Jul 2025, Lin et al., 6 May 2025).

This approach is robust and unifies the analysis across all dimensions, bypassing the limitations of prior methods tailored to one or two dimensions (e.g., white noise decompositions, orthogonal polynomial expansions).

The technical workflow is as follows:

• Decompose the field: $X(x) = \sum_{j=0}^\infty Y_j(x)$, with $Y_j$ smooth and independent.
• Approximate the chaos measure at scale $m$ as $I_m(dt) = \prod_{j=0}^m P_j(t) dt$ (with $P_j$ depending on $Y_j$).
• For multi-scale dyadic cubes, obtain local Fourier bounds by integration by parts and regularity of $Y_j$.
• Apply vector-valued martingale inequalities to sum the local bounds, yielding a sharp global rate for the Fourier coefficients.

5. Martingale Methods, Uniqueness, and Absolute Continuity

The martingale approach, both in the $T$-martingale sense and via vector-valued martingales, is fundamental. It underlies:

• The almost sure convergence of approximating measures (martingale limit in $L^1$), ensuring unique definition for the chaos measure.
• The independence from the specific regularization scheme, as demonstrated in uniqueness results for GMC measures (Shamov, 2014, Junnila et al., 2015).
• The transfer of Fourier dimension results under mutual absolute continuity (e.g., for non-Gaussian log-correlated Fourier series coefficients, via coupling arguments and invariance principles) (Kim et al., 25 Oct 2024, Chowdhury et al., 24 Feb 2025), showing universality of Fourier dimension phenomena beyond the Gaussian setting.

6. Connections to Multifractal Analysis and Spectral Properties

The computed Fourier dimension equals the correlation dimension: in particular,

$$\mathrm{dim}_F(M_\gamma) = \mathrm{dim}_2(M_\gamma)$$

where $\mathrm{dim}_2$ is derived from the scaling of second moments (energy integrals). This equivalence has deep implications:

• Multifractal (scaling) analysis for the measure is reflected in its harmonic analytic characteristics.
• The GMC measure is “Rajchman,” i.e., its Fourier coefficients decay to zero, and the polynomial decay aligns with its thick-point structure (Garban et al., 2023).
• Applications range from Liouville quantum gravity (random geometry), conformal field theory (via tachyon fields and KPZ relations), to the spectral analysis of random fractal measures.

7. Broader Implications, Universality, and Further Directions

The elucidation of exact Fourier dimension for GMC on $\mathbb{T}^d$:

• Provides an explicit spectral fingerprint for random measures in probabilistic models of turbulence, quantum gravity, and random matrix theory (Najnudel et al., 20 Feb 2025).
• Facilitates construction of random Salem sets and informs projection theorems in fractal geometry (Chen et al., 20 Sep 2024, Lin et al., 6 May 2025).
• Demonstrates the robustness of multifractal and harmonic regularity under perturbations such as non-Gaussian disorder or generalized cascade models, via invariance principles and absolute continuity (Kim et al., 25 Oct 2024, Chowdhury et al., 24 Feb 2025).
• Suggests new tools (integration by parts for regularized fields, vector-valued martingale analysis) for higher-dimensional and more general settings in random measure theory.

These developments synthesize probabilistic, analytic, and geometric insights to fully describe the harmonic structure of Gaussian multiplicative chaos and serve as a template for deeper paper of multifractal random measures.