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Harmonic analysis of multiplicative chaos Part I: the proof of Garban-Vargas conjecture for 1D GMC

Zhaofeng Lin, Yanqi Qiu, Mingjie Tan

TL;DR

This work completes Part I of a program on the harmonic analysis of Gaussian multiplicative chaos by proving the Garban-Vargas conjecture in 1D. It develops a refined vector-valued martingale method with Bacry-Muzy white-noise decomposition to achieve sharp, almost-sure polynomial decay of GMC Fourier coefficients, establishing the exact Fourier dimension $D_mma$ for all sub-critical $mma$. The authors combine dyadic Littlewood-Paley-type localization, Abel summation, and Hölder regularity to derive a uniform $L^p(ell^q)$ bound for the associated vector-valued martingale, which yields the lower bound $ ext{dim}_F=mbda$. The upper bound follows from a classical potential-theoretic control linking the Fourier and correlation dimensions and from the GMC $L^2$-spectrum, confirming that the Fourier dimension matches the correlation dimension for the 1D sub-critical GMC and enabling a Fourier-restriction framework for associated random measures.

Abstract

In this paper, we establish the exact Fourier dimensions of all standard sub-critical Gaussian multiplicative chaos on the unit interval, thereby confirming the Garban-Vargas conjecture. The proof relies on a significant improvement of the vector-valued martingale method, initially developed by Chen-Han-Qiu-Wang in the studies of the Fourier dimensions of Mandelbrot cascade random measures.

Harmonic analysis of multiplicative chaos Part I: the proof of Garban-Vargas conjecture for 1D GMC

TL;DR

This work completes Part I of a program on the harmonic analysis of Gaussian multiplicative chaos by proving the Garban-Vargas conjecture in 1D. It develops a refined vector-valued martingale method with Bacry-Muzy white-noise decomposition to achieve sharp, almost-sure polynomial decay of GMC Fourier coefficients, establishing the exact Fourier dimension for all sub-critical . The authors combine dyadic Littlewood-Paley-type localization, Abel summation, and Hölder regularity to derive a uniform bound for the associated vector-valued martingale, which yields the lower bound . The upper bound follows from a classical potential-theoretic control linking the Fourier and correlation dimensions and from the GMC -spectrum, confirming that the Fourier dimension matches the correlation dimension for the 1D sub-critical GMC and enabling a Fourier-restriction framework for associated random measures.

Abstract

In this paper, we establish the exact Fourier dimensions of all standard sub-critical Gaussian multiplicative chaos on the unit interval, thereby confirming the Garban-Vargas conjecture. The proof relies on a significant improvement of the vector-valued martingale method, initially developed by Chen-Han-Qiu-Wang in the studies of the Fourier dimensions of Mandelbrot cascade random measures.

Paper Structure

This paper contains 26 sections, 16 theorems, 202 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background on GMC
  3. Main result
  4. Frostman regularity and Fourier restriction estimate
  5. Outline of the proof of Theorem \ref{['FourierDim-GMC']}
  6. The upper bound
  7. The lower bound via vector-valued martingale method
  8. Preliminaries
  9. Fourier dimension and correlation dimension
  10. Martingale type $p$ inequalities for $\ell^q$
  11. Construction of GMC via Bacry-Muzy's white noise decomposition
  12. Initial steps in the proof of Theorem \ref{['Uniform-Bound']}
  13. An elementary identity
  14. A very rough estimate of $\widehat{\mu_{\gamma, m}}$
  15. Existence of suitable exponents for $L^p(\ell^q)$
  16. ...and 11 more sections

Key Result

Theorem 1.1

For each $\gamma\in(0,\sqrt{2})$, almost surely, we have $\mathrm{dim}_{F}(\mu_{\gamma,\mathrm{GMC}})=D_{\gamma}$.

Figures (4)

  • Figure 1: The sets $\mathcal{C}_m(t)$ (left) and $A_m(t)$ (right).
  • Figure 2: $|t_1-t_2|\geq2^{-(m-1)} \Longrightarrow A_m(t_1)$ and $A_m(t_2)$ are disjoint.
  • Figure 3: $\mathrm{dist}(S,T)\geq2^{-(m-1)}\Longrightarrow \bigcup\limits_{t_1\in T}A_m(t_1)\cap \bigcup\limits_{t_2\in S}A_m(t_2) = \varnothing$.
  • Figure 4: The regions corresponding to $\xi_{\mathrm{left}}(t), \xi_{\mathrm{right}}(t), \xi_{\mathrm{center}}(t)$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • ...and 22 more