Holomorphic Multiplicative Chaos (HMC)

• Holomorphic Multiplicative Chaos is defined as limits of exponentiating log‐correlated holomorphic fields, capturing multifractal scaling and universality in random matrices and number theory.
• It reveals complex Fourier coefficient limits that manifest sharp phase transitions and polynomial decay in both subcritical and critical regimes.
• Advanced analytical tools such as Gaussian approximation, SDE limits, and Malliavin calculus underpin its convergence proofs and connections to quantum gravity and Liouville theory.

Holomorphic Multiplicative Chaos (HMC) is a class of random measures emerging as limits of exponentials of log-correlated (often holomorphic or complex Gaussian) fields, generalizing the theory of Gaussian multiplicative chaos (GMC) to holomorphic settings. HMC arises naturally in the paper of characteristic polynomials of large random matrices, most notably in the Circular Unitary Ensemble (CUE) and the circular-β-ensemble, and encodes multifractal behavior, universality, and intricate scaling properties. The stochastic models and convergence proofs in this domain illuminate deep connections between random matrix theory, analytic number theory, fractal geometry, and harmonic analysis.

1. Foundational Construction and Gaussian Approximation

HMC is defined as a limit of random measures built by exponentiating holomorphic (or log-correlated complex Gaussian) fields. For random matrix models, this frequently takes the form

$$\mathrm{HMC}_\theta(z) = \exp\left(\sqrt{\theta}\ G^\mathbb{C}(z)\right)$$

where $$G^\mathbb{C}(z) = \sum_{k=1}^\infty \frac{z^k}{\sqrt{k}}\mathcal{N}_k$$, with $\mathcal{N}_k$ independent complex Gaussians, and $\theta$ is a parameter (often $2/\beta$ for circular-β ensembles).

A critical technical ingredient is the "good Gaussian approximation" of the underlying field—non-Gaussian structures (as observed, for instance, in stochastic models for the Riemann zeta function (Saksman et al., 2016)) can be approximated uniformly by Gaussian fields with matching covariance up to a bounded error: $X_N(x) = G_N(x) + E_N(x)$ with $G_N(x)$ a Gaussian field with logarithmic covariance, and $E_N(x)$ a uniformly convergent error possessing finite exponential moments. This decoupling enables the direct application of GMC theory, ensuring the transfer of multifractal properties from the Gaussian case to the non-Gaussian (and holomorphic) multiplicative chaos regimes.

2. Fourier (Secular) Coefficient Limits and Multifractality

A central object of paper in HMC is the limiting behavior of its Fourier coefficients

$$c_n = [z^n]\exp\left(\sqrt{\theta}\sum_{k=1}^{\infty}\frac{z^k}{\sqrt{k}}\mathcal{N}_k\right)$$

and their distributional convergence in various "phases" (critical, subcritical). In the subcritical $L^1$-phase ($0 < \theta < 1$), normalization by the second moment yields convergence in distribution: $$\frac{c_n}{\sqrt{\mathbb{E}|c_n|^2}} \xrightarrow{d} \sqrt{\mathcal{M}_\theta}\mathcal{Z}$$ where $$\mathcal{M}_\theta = (1/2\pi)\int_0^{2\pi} \mathrm{GMC}_\theta(d\phi)$$ is the total mass of the associated GMC on the circle, and $\mathcal{Z}$ is standard complex normal (Najnudel et al., 20 Feb 2025). In the critical phase ($\theta = 1, \beta = 2$), rescaling by $(\log n)^{1/4}$ instead produces the analogous scaling limit, revealing "better than square-root cancellation" (Atherfold et al., 19 Aug 2025): $$c_n \cdot (\log n)^{1/4} \xrightarrow{d} \sqrt{\mathcal{M}_1}\mathcal{Z}$$ with explicit laws for $\mathcal{M}_1$ (the inverse having exponential distribution with mean $\sqrt{\pi}$), and joint convergence of blocks of consecutive coefficients to Gaussian vectors modulated by Toeplitz matrices built from the GMC mass.

This multifractal structure manifests in the scaling exponents of fractional moments: $$\mathbb{E}[|c_n|^{2q}] \asymp (\log n)^{-q/2},\quad q \in (0,1)$$ which holds in both Gaussian and universal non-Gaussian regimes with sufficiently light tails (Gu et al., 11 Jan 2024). The moments and their combinatorial representations (e.g., magic square formulas) connect HMC to Jack function theory, random permutations, and Tauberian estimates (Najnudel et al., 2020).

3. Convergence Criteria, Phase Transitions, and Universality

The convergence and universality of HMC measures and coefficients are governed by tail properties of the underlying field. For i.i.d. rotationally invariant complex coefficients $X_k$ in the generating series, the scaling of low moments persists universally under sub-exponential moment conditions ($\mathbb{E}[e^{\gamma|X_k|}] < \infty$ for $\gamma > 2q$) (Gu et al., 11 Jan 2024). In contrast, fields with stretched exponential tails exhibit qualitatively different behavior, with emergent double-layer phase transitions around the critical tail.

Phase transitions also arise in the dependence of the Fourier/Sobolev regularity of the chaos: HMC almost surely belongs to $H^s$ spaces for any $s$ below a sharp threshold ($s_\theta = -\theta/2$ in subcritical/critical, $s_\theta = -\sqrt{\theta} + 1/2$ in supercritical) (Najnudel et al., 2020).

4. Harmonic Analysis and Fourier Dimension

The harmonic analytic properties of HMC, specifically the decay of its Fourier transform and the determination of its Fourier dimension, reflect its underlying multifractal geometry. Recent unified approaches using vector-valued martingale inequalities and b-adic partitions yield sharp polynomial decay estimates for Fourier transforms, confirming conjectures such as Garban-Vargas for sub-critical GMC (Lin et al., 6 May 2025). The Fourier dimension can often be computed explicitly and matches the correlation (energy) dimension in canonical models, with similar methodology expected to apply to holomorphic chaos via dyadic approximations and Hölder-moment analysis.

Canonical Mandelbrot coverings and cascades, as well as Poisson chaos, fit into the martingale framework that links these harmonic properties to the multifractal nature of HMC. In the holomorphic setting, such polynomial decay reveals the roughness and frequency-space complexity encoded by the chaos measure, especially in its critical phase.

5. Connections to Random Matrices, Number Theory, and Related Models

HMC inherently arises from the paper of characteristic polynomials of large random unitary matrices (CUE, circular-β-ensemble), and its secular coefficient limits encode the fluctuations of these polynomials (Najnudel et al., 20 Feb 2025, Atherfold et al., 19 Aug 2025). In the subcritical and critical phases ($\beta > 2$ and $\beta = 2$ respectively), the law of thick points and extreme fluctuations match precisely those of GMC, confirming universality conjectures and providing explicit asymptotics for extreme events and maxima (Junnila et al., 2022, Chhaibi et al., 2019).

Analogous structures appear in number theory: stochastic models for the Riemann zeta function (e.g., random Euler products) constructed via finite sums over primes with random phases converge, after normalization, to non-Gaussian chaos measures approximable by GMC (Saksman et al., 2016). The bridge between GMC and non-Gaussian chaos through “freezing” techniques opens rigorous avenues for studying HMC within analytic number theory, Liouville theory, and even financial models of rough volatility and multifractal random walks (Hager et al., 2020).

6. Analytical and Probabilistic Tools: SDE Limits, Martingale CLTs, and Malliavin Calculus

Advanced spectral and probabilistic techniques are central in the paper of HMC. Stochastic differential equation (SDE) limits for random orthogonal polynomials (as in the circular-β-ensemble) exactly describe the limiting diffusion of the squared modulus process, and moment calculations rely on Doob-Meyer decompositions, Itô calculus, and Dufresne’s identity for exponential functionals (Chhaibi et al., 2019).

Martingale central limit theorems underpin process convergence for blocks of secular coefficients, and Tauberian arguments govern asymptotic control. The use of Malliavin calculus—recently extended to GMC inverses and suggested to apply to HMC—enables derivation of integration by parts formulas and analysis of quantile and inverse functions (Kojar, 2023). These analytical tools are critical for both fine probabilistic estimates and geometric interpretations (e.g., in conformal welding and quantum gravity contexts).

7. Open Problems and Future Directions

Outstanding problems in HMC revolve around the precise geometric structure of the chaos measure (especially correlations and dimensionality), non-Gaussian generalizations, the behavior under different scaling regimes of secular indices (beyond sublinear), and extensions to Liouville-type quantum gravity and conformal field theory.

Other directions involve the construction of explicit chaos models from thick points of underlying log-correlated fields, universality beyond the Gaussian setting (particularly with heavy-tailed fields), and the analytic characterization of phase transitions. The interplay between random matrices, combinatorial objects (magic squares, Jack functions), and complex analytic fields provides further avenues for exploration.

Summary Table: Fundamental Features of HMC

Aspect	Description	Key Reference(s)
Field Construction	Exponentiate log-correlated holomorphic field	(Najnudel et al., 20 Feb 2025, Saksman et al., 2016)
Fourier Coefficient Law	$$c_n(\log n)^{1/4} \to \sqrt{\mathcal{M}_1}\mathcal{Z}$$ (critical)	(Atherfold et al., 19 Aug 2025)
Multifractality	Fractional moment scaling, multifractal spectrum	(Najnudel et al., 2020, Gu et al., 11 Jan 2024)
Fourier Dimension	Sharp polynomial decay (dim matches correlation)	(Lin et al., 6 May 2025)
Matrix Model Link	Secular coeff. converge to HMC coefficients	(Najnudel et al., 20 Feb 2025, Atherfold et al., 19 Aug 2025)
Universality	Phase transitions, Gaussian and non-Gaussian	(Gu et al., 11 Jan 2024, Aggarwal et al., 2021)

Holomorphic Multiplicative Chaos thus serves as a unifying paradigm for understanding multifractal and universal features of exponentiated complex log-correlated fields, linking rigorous analysis in probability, random matrix theory, and number theory. Further research continues to clarify its geometric, combinatorial, and spectral characteristics.