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Invariance principle for the Gaussian Multiplicative Chaos via a high dimensional CLT with low rank increments

Mriganka Basu Roy Chowdhury, Shirshendu Ganguly

TL;DR

This work establishes a universality principle for Gaussian multiplicative chaos (GMC) in the subcritical regime by coupling Gaussian and non-Gaussian log-correlated noises. The authors prove that GMC measures $\mu_{\gamma,g}$ and $\mu_{\gamma,a}$ are mutually absolutely continuous for all $\gamma\in(0,\sqrt{2})$ via a novel high-dimensional central limit theorem for rank-one increments, implemented through Skorokhod embeddings and a new perturbation bound for square roots of positive semidefinite matrices. A key technical advance is a covariance-sensitive CLT whose error depends on the spectral norm of $U=\frac{1}{n}\sum_i v_i v_i^T$ and on the isotropy provided by Fourier-analytic structure, enabling a hierarchical, thick-point reduction that makes the comparison tractable. The results extend to higher dimensions and suggest a broadly applicable framework for universality in log-correlated systems beyond GMC, with potential implications for quantum field theories and Liouville quantum gravity.

Abstract

Gaussian multiplicative chaos (GMC) is a canonical random fractal measure obtained by exponentiating log-correlated Gaussian processes, first constructed in the seminal work of Kahane (1985). Since then it has served as an important building block in constructions of quantum field theories and Liouville quantum gravity. However, in many natural settings, non-Gaussian log-correlated processes arise. In this paper, we investigate the universality of GMC through an invariance principle. We consider the model of a random Fourier series, a process known to be log-correlated. While the Gaussian Fourier series has been a classical object of study, recently, the non-Gaussian counterpart was investigated and the associated multiplicative chaos constructed by Junnila in 2016. We show that the Gaussian and non-Gaussian variables can be coupled so that the associated chaos measures are almost surely mutually absolutely continuous throughout the entire sub-critical regime. This solves the main open problem from Kim and Kriechbaum (2024) who had earlier established such a result for a part of the regime. The main ingredient is a new high dimensional CLT for a sum of independent (but not i.i.d.) random vectors belonging to rank one subspaces with error bounds involving the isotropic properties of the covariance matrix of the sum, which we expect will find other applications. The proof relies on a path-wise analysis of Skorokhod embeddings as well as a perturbative result about square roots of positive semi-definite matrices which, surprisingly, appears to be new.

Invariance principle for the Gaussian Multiplicative Chaos via a high dimensional CLT with low rank increments

TL;DR

This work establishes a universality principle for Gaussian multiplicative chaos (GMC) in the subcritical regime by coupling Gaussian and non-Gaussian log-correlated noises. The authors prove that GMC measures and are mutually absolutely continuous for all via a novel high-dimensional central limit theorem for rank-one increments, implemented through Skorokhod embeddings and a new perturbation bound for square roots of positive semidefinite matrices. A key technical advance is a covariance-sensitive CLT whose error depends on the spectral norm of and on the isotropy provided by Fourier-analytic structure, enabling a hierarchical, thick-point reduction that makes the comparison tractable. The results extend to higher dimensions and suggest a broadly applicable framework for universality in log-correlated systems beyond GMC, with potential implications for quantum field theories and Liouville quantum gravity.

Abstract

Gaussian multiplicative chaos (GMC) is a canonical random fractal measure obtained by exponentiating log-correlated Gaussian processes, first constructed in the seminal work of Kahane (1985). Since then it has served as an important building block in constructions of quantum field theories and Liouville quantum gravity. However, in many natural settings, non-Gaussian log-correlated processes arise. In this paper, we investigate the universality of GMC through an invariance principle. We consider the model of a random Fourier series, a process known to be log-correlated. While the Gaussian Fourier series has been a classical object of study, recently, the non-Gaussian counterpart was investigated and the associated multiplicative chaos constructed by Junnila in 2016. We show that the Gaussian and non-Gaussian variables can be coupled so that the associated chaos measures are almost surely mutually absolutely continuous throughout the entire sub-critical regime. This solves the main open problem from Kim and Kriechbaum (2024) who had earlier established such a result for a part of the regime. The main ingredient is a new high dimensional CLT for a sum of independent (but not i.i.d.) random vectors belonging to rank one subspaces with error bounds involving the isotropic properties of the covariance matrix of the sum, which we expect will find other applications. The proof relies on a path-wise analysis of Skorokhod embeddings as well as a perturbative result about square roots of positive semi-definite matrices which, surprisingly, appears to be new.

Paper Structure

This paper contains 18 sections, 14 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Gaussian multiplicative chaos (GMC)
  3. High dimensional CLT
  4. Idea of the proof
  5. Organization
  6. Acknowledgements
  7. High dimensional CLT: Proof of Theorem \ref{['thm:main']}
  8. Construction of the coupling.
  9. Bounding the error per coordinate
  10. Perturbations of square roots of positive semi-definite matrices
  11. Bounding the operator norm of $A_s$
  12. Bounding the quadratic variation
  13. Application to the multiplicative chaos: Proof of Theorem \ref{['thm:gmc']}
  14. The hierarchical model
  15. Spectral norm bounds using the Fourier structure
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\gamma \in (0, \sqrt 2)$. Then, there is a coupling of $\left\{a^{(1)}_i, a^{(2)}_i\right\}_{i \geqslant 1}$ and $\left\{g^{(1)}_i, g^{(2)}_i\right\}_{i \geqslant 1}$ such that that is, the multiplicative chaos measures are mutually absolutely continuous.

Figures (1)

  • Figure : Figure. Plots of the (prelimiting) GMC density associated with the random Fourier series $[0, 1] \ni t \mapsto \sum_{k = 1}^n k^{-1/2} \left(g_k^{(1)} \cos(2\pi kt) + g_k^{(2)}\sin(2\pi kt)\right), \quad g_k^{(1)}, g_k^{(2)} \sim \mathcal{N}(0, 1).$The six plots show the densities for varying values of the number of terms $n$ (varied along columns), and intermittency parameter $\gamma$ (varied along rows). The density is proportional to the exponential of ($\gamma$ times) the Fourier series, properly normalized. The GMC for a fixed $\gamma$ is the large-$n$ limit of the measures with these densities. Note that the measure is much rougher and the support much smaller for larger values of $\gamma < \sqrt{2}$. For $\gamma \geqslant \sqrt{2}$, the support vanishes in the $n \to \infty$ limit, and thus the GMC is trivial, as illustrated in the plots corresponding to $\gamma = 1.9$ (last column).

Theorems & Definitions (28)

  • Theorem 1.1: Universality of GMC
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2: Square-root gap
  • Remark 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Lemma \ref{['lem:sqrtgap']}
  • Lemma 2.5: Matrix Bernstein inequality
  • ...and 18 more