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Harmonic analysis of Gaussian multiplicative chaos on the circle

Christophe Garban, Vincent Vargas

Abstract

In this paper, we initiate the harmonic analysis of Gaussian multiplicative chaos (GMC) on the circle, i.e. the study of its Fourier coefficients. In particular, we show that almost surely GMC is a so-called Rajchman measure which means that its Fourier coefficients converge to $0$ when the frequency goes to infinity. We supplement this result with a convergence in law result for the rescaled Fourier coefficients.

Harmonic analysis of Gaussian multiplicative chaos on the circle

Abstract

In this paper, we initiate the harmonic analysis of Gaussian multiplicative chaos (GMC) on the circle, i.e. the study of its Fourier coefficients. In particular, we show that almost surely GMC is a so-called Rajchman measure which means that its Fourier coefficients converge to when the frequency goes to infinity. We supplement this result with a convergence in law result for the rescaled Fourier coefficients.
Paper Structure (11 sections, 8 theorems, 81 equations)

This paper contains 11 sections, 8 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Introduction and state of the art.
  3. The main results and perspectives
  4. Main results.
  5. Perspectives and conjectures.
  6. Relation to other models and results.
  7. Proof of the Rajchman property
  8. Proof of Theorem \ref{['RajchmanbackL4']} via $L^4$ moments
  9. Convergence in law (Theorem \ref{['thlaw']})
  10. Convergence in law for a "toy-model" on $[0,1]$.
  11. The case of the GMC on the unit circle.

Key Result

Theorem 1.1

For all $\gamma< \sqrt{2}$, the sequence $c_n$ goes to $0$ almost surely as $n$ goes to infinity.

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2
  • Remark 3
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Remark 4
  • ...and 4 more