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Derivatives of Gaussian multiplicative chaos

Antoine Jego

Abstract

Consider a logarithmically-correlated Gaussian field $X$ in $d$ dimensions. For all $γ\in (-\sqrt{2d},\sqrt{2d})$, we show that the derivatives $\frac{\partial^k}{\partialγ^k} :e^{γX_ε}:$ of the regularised Gaussian multiplicative chaos $:e^{γX_ε}:$ converge as $ε\to 0$. By deriving optimal bounds on their growth as $k\to\infty$, we control the power expansion of $:e^{γX_ε}:$ about each $γ\in(-\sqrt{2d},\sqrt{2d})$. This yields an alternative approach to complex Gaussian multiplicative chaos in the whole subcritical regime, based entirely on real-valued quantities. One of our key technical contributions is to provide a truncated second moment approach to the uniform integrability of the derivatives of multiplicative chaos and its associated complex variant.

Derivatives of Gaussian multiplicative chaos

Abstract

Consider a logarithmically-correlated Gaussian field in dimensions. For all , we show that the derivatives of the regularised Gaussian multiplicative chaos converge as . By deriving optimal bounds on their growth as , we control the power expansion of about each . This yields an alternative approach to complex Gaussian multiplicative chaos in the whole subcritical regime, based entirely on real-valued quantities. One of our key technical contributions is to provide a truncated second moment approach to the uniform integrability of the derivatives of multiplicative chaos and its associated complex variant.
Paper Structure (16 sections, 18 theorems, 154 equations, 1 figure)

This paper contains 16 sections, 18 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Context
  4. Strategy
  5. Organisation
  6. Setup
  7. Preliminaries
  8. Hermite polynomials
  9. Proof of Theorem \ref{['T:main']} when gamma=0
  10. Basic properties of almost star-scale invariant fields
  11. Two elementary lemmas
  12. Proof of Theorem \ref{['T:convergence_k']}
  13. Second moment, good event
  14. Higher moment, good event
  15. First moment, bad event
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\gamma \in (-\sqrt{2d},\sqrt{2d})$ and $f:D \to \mathbb{R}$ be a bounded measurable function compactly supported in $D$. For all $k \ge 0$, the following limit exists in $L^1$: Moreover, for all $\gamma' \in \mathbb{C}$ with $|\gamma'-\gamma|<\sqrt{d}-|\gamma|/\sqrt{2}$, the sequence converges in $L^1$ to

Figures (1)

  • Figure 1: Illustration of the eye-shaped domain $\mathsf{Eye}$\ref{['E:eye']} and a disc centred at some point $\gamma \in (-\sqrt{2d},\sqrt{2d})$ with radius $\sqrt{d}-|\gamma|/\sqrt2$. As $\gamma$ varies in $(-\sqrt{2d},\sqrt{2d})$, these discs cover all the set $\mathsf{Eye}$. In this paper, we define $:\!e^{\gamma'X}\!:$ in the whole set $\mathsf{Eye}$ by defining it in each such discs.

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3: Brownian multiplicative chaos
  • Remark 1.4: Construction of complex GMC
  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 30 more