On the Fourier coefficients of critical Gaussian multiplicative chaos

This paper establishes that the Fourier coefficients of critical Gaussian multiplicative chaos on the unit interval decay faster than $(\log n)^{-\alpha}$ in probability for any $\alpha < 1/4$, thereby extending recent work by Garban and Vargas.

The Gist

The Big Picture: A Stormy Sea of Numbers

Imagine you are standing on a beach looking at the ocean. The water isn't calm; it's a chaotic, churning mess of waves, foam, and spray. In mathematics, this "stormy sea" is called Gaussian Multiplicative Chaos (GMC).

Think of GMC as a way to describe a surface that is infinitely rough. If you zoom in on any tiny spot, it looks just as jagged as the whole thing. It's like a fractal coastline that never smooths out, no matter how close you look.

Mathematicians use this to model things in the real world that are wildly unpredictable, like:

• Turbulent wind in a hurricane.
• The erratic movement of stock prices.
• The distribution of mass in the early universe.

The "Critical" Moment: The Edge of Chaos

The paper focuses on a specific, very delicate state of this chaos called the "Critical" point.

Imagine a pot of water on a stove.

• Subcritical (Cool): The water is calm. You can easily predict the ripples.
• Supercritical (Boiling): The water is a violent boil. It's chaotic, but the energy is so high that the chaos is "loud" and obvious.
• Critical (The Edge): This is the exact moment the water is about to boil. It's a state of perfect tension. The waves are huge, but they are also incredibly fragile. This is the hardest state to study because it's right on the knife-edge between order and total disorder.

The authors are studying the "Fourier Coefficients" of this critical state.

What are Fourier Coefficients? (The Musical Analogy)

To understand what the authors are measuring, imagine the chaotic ocean surface is a piece of music.

• A Fourier Coefficient is like asking: "How much of a specific musical note (frequency) is in this sound?"
• Low frequencies are the deep, rumbling bass notes (big, slow waves).
• High frequencies are the high-pitched, rapid whistles (tiny, fast ripples).

The question the paper asks is: As we listen to higher and higher notes (higher frequencies), does the sound of this chaotic ocean fade away to silence?

In math, if the sound fades away, the measure is called a Rajchman measure. It's a fancy way of saying, "The chaos smooths out when you look at it from a distance."

The Main Discovery: A Slow Fade

For a long time, mathematicians knew that if the chaos wasn't too intense (subcritical), the high notes faded away quickly. But for the Critical chaos (the edge of the pot), no one knew for sure if the sound would ever stop.

The authors proved that yes, the sound does fade away, but it does so very, very slowly.

They found that the volume of the high notes drops off like a "logarithmic" decay.

• Analogy: Imagine a lightbulb that is supposed to turn off. In a normal room, it clicks off instantly. In this critical chaos, the lightbulb is dimming so slowly that it takes an incredibly long time to go dark.
• The Math: They proved that if you multiply the volume of the note by a specific factor (related to the logarithm of the note's pitch), the result eventually goes to zero.

How Did They Prove It? (The "Good Day" Strategy)

Proving this is hard because the "ocean" is so wild that standard math tools break. The authors had to be clever.

1. The Problem: If you try to measure the whole ocean at once, the waves are too big and unpredictable. The math explodes.
2. The Solution: They decided to only measure the ocean on "Good Days."
   • They defined a "Good Day" as a scenario where the waves don't get too crazy, even though they are still chaotic.
   • They proved that "Bad Days" (where the waves are infinitely huge) happen so rarely that they don't matter for the final result.
3. The Oscillation Trick: The high-frequency notes (the Fourier coefficients) act like a rapidly vibrating string. When you multiply a chaotic wave by a rapidly vibrating string, they often cancel each other out (like noise-canceling headphones).
   • The authors had to show that this "cancellation" happens often enough, even in the critical state, to make the high notes disappear.

Why Does This Matter?

This paper solves a puzzle that has been open for years.

• Before: We knew critical chaos was weird and singular (it lives on a set of points so small they have zero area, yet it holds all the mass). We didn't know if it had a "smooth" side.
• Now: We know it does have a smooth side. Even though the chaos is extreme, if you listen to the high-pitched frequencies, the signal eventually dies out.

Summary in One Sentence

The authors proved that even in the most extreme, "edge-of-boiling" state of mathematical chaos, the high-frequency ripples eventually fade into silence, but they do so at a glacial, logarithmic pace.

Technical Summary

1. Problem Statement and Context

Background:
Gaussian Multiplicative Chaos (GMC) is a mathematical theory initiated by Kahane to rigorously define measures of the form $\mu_\gamma(dx) = e^{\gamma X(x) - \frac{\gamma^2}{2}\mathbb{E}[X(x)^2]}dx$, where $X$ is a log-correlated Gaussian field.

• Subcritical Phase ($\gamma < \sqrt{2}$): The measure is non-degenerate and well-understood. Recent work by Garban and Vargas established that for $\gamma < \sqrt{2}$, the Fourier coefficients $\hat{\mu}(n)$ decay to zero (the measure is a Rajchman measure), and they conjectured specific decay rates related to the Fourier dimension.
• Critical Phase ($\gamma = \sqrt{2}$): This is the boundary case where the standard exponential definition fails (the measure becomes zero). A non-trivial limit is obtained via the derivative martingale or Seneta-Heyde normalization.
  • It is known that the Hausdorff dimension of critical GMC is 0.
  • Since the Fourier dimension is bounded above by the Hausdorff dimension, the Fourier dimension of critical GMC is 0.
  • The Open Problem: Despite having Fourier dimension 0, it was unknown whether the Fourier coefficients $\hat{\mu}(n)$ actually vanish as $n \to \infty$ (i.e., whether critical GMC is a Rajchman measure).

The Specific Question:
Do the Fourier coefficients $c_n$ of the critical GMC measure on the unit interval converge to zero in probability as $n \to \infty$? If so, at what rate?

2. Methodology

The authors employ a probabilistic approach combining second-moment estimates, change of measure techniques (Girsanov's theorem), and Brownian motion barrier arguments.

Key Technical Steps:

1. Approximation and Truncation:
   The critical measure $\mu$ is defined as the limit of approximations $\mu_t$. The authors analyze the $n$-th Fourier coefficient of the approximation, $c_{n,t} = \int e^{in\theta} \mu_t(d\theta)$. They aim to bound $\mathbb{E}[|c_{n,t}|^2]$ on a "good" event.

2. The "Good" Event ($E_{n,t}(A)$):
   To control the second moment, the authors restrict the integration to pairs of points $(\theta_1, \theta_2)$ where the underlying Gaussian field $X_s(\theta)$ behaves "typically" (does not fluctuate too wildly).

   • They define a barrier function $U(s) \approx \sqrt{2s} - \frac{1}{2\sqrt{2}}\log s$.
   • The event $E_{n,t}(A)$ requires that for all $\theta$, the field $X_s(\theta)$ stays below $U(s) + A$ for scales $s$ ranging from a frequency-dependent cutoff $r_n = \delta \log n$ up to $t$.
   • This restriction is crucial because the standard second moment diverges at criticality without such constraints.

3. Decomposition of the Integral:
   The expectation $\mathbb{E}[|c_{n,t}|^2]$ involves a double integral over $\theta_1, \theta_2$. The authors split the domain based on the distance $\Delta = |\theta_2 - \theta_1|$:

   • Oscillatory Range ($|\Delta| \ge \Delta_n = e^{-r_n}$): Here, the term $e^{in\Delta}$ oscillates rapidly. The authors use integration by parts to exploit this cancellation. They transform the expectation using a tilted measure (Girsanov) to handle the exponential weights, showing this contribution is negligible ($O((\log n)^{-2})$).
   • Dominant Range ($|\Delta| < \Delta_n$): Here, the oscillation is slow. The authors ignore the oscillation (using the triangle inequality) and focus on bounding the mass of the measure in this small region.

4. Brownian Barrier Estimates:
   The core of the proof involves estimating the probability that two correlated Brownian motions (representing the field at $\theta_1$ and $\theta_2$) stay below the barrier $U(s)$ up to time $t$.

   • They utilize the branching time $r(\Delta) = \log(1/|\Delta|)$, which is the time scale at which the two paths decouple.
   • They derive precise upper bounds for the joint probability of these paths staying under the barrier, utilizing reflection principles and estimates for Brownian bridges.
   • A key insight is that the dominant contribution comes from scales where the "branching time" is close to the cutoff $r_n$, leading to a specific logarithmic scaling.

3. Key Contributions and Results

Main Theorem (Theorem 1.1):
Let $\mu$ be the critical GMC measure on $[0,1]$ and $c_n$ its $n$-th Fourier coefficient. For any $\alpha < 1/4$, the sequence $(\log n)^\alpha c_n$ converges to zero in probability as $n \to \infty$.

• Implication: This confirms that critical GMC is indeed a Rajchman measure (its Fourier coefficients vanish at infinity).
• Rate of Decay: The result establishes a polylogarithmic decay rate. Specifically, $|c_n|$ decays roughly as $(\log n)^{-1/4}$ in probability.

Conjectures and Heuristics:

• The authors provide heuristics suggesting the true order of magnitude is likely closer to $(\log n)^{-1}$.
• They propose Conjecture 1.4: For the critical case ($\gamma = 1/\sqrt{2}$ in their normalized notation, corresponding to $\gamma=\sqrt{2}$ in standard notation), the quantity $(\log n)^{1/4} n^{1/4} |c_n|$ converges in distribution to a non-trivial random variable.
  • Note: The $n^{1/4}$ factor relates to the Fourier dimension of the subcritical case, while the $(\log n)^{1/4}$ factor arises from the "freezing" phenomenon at criticality.

4. Significance

1. Resolution of a Longstanding Open Problem: The paper answers the primary open question left by Garban and Vargas (2020) regarding the Rajchman property of critical GMC. It proves that despite having zero Hausdorff dimension, the measure still exhibits Fourier decay.
2. Understanding Critical Phenomena: The work sheds light on the "freezing" phenomenon in multiplicative chaos. At criticality, the measure is supported on a set of dimension zero, yet the Fourier coefficients do not vanish instantly; they decay very slowly (polylogarithmically).
3. Methodological Innovation: The paper introduces a refined technique for handling the second moment of Fourier coefficients in the critical regime. By restricting the analysis to "good" points only at scales $t \ge r_n$ (rather than all scales), they manage to control the oscillating integral, a technique that overcomes the breakdown of standard $L^2$ methods at criticality.
4. Connection to Other Fields: The results have implications for the study of extreme values of log-correlated fields, random matrix theory, and the theory of Salem sets, providing a more complete picture of the harmonic analysis of multifractal measures.

In summary, Arguin and Hamdan prove that critical Gaussian Multiplicative Chaos is a Rajchman measure with a specific polylogarithmic decay rate for its Fourier coefficients, bridging a significant gap in the understanding of critical random measures.