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The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency

Joseph Najnudel, Elliot Paquette, Nick Simm, Truong Vu

TL;DR

This work analyzes the Fourier coefficients of holomorphic multiplicative chaos (HMC) in the $L^{1}$-phase ($0<\theta<1$), showing that normalized coefficients converge in law to a complex Gaussian scaled by the GMC mass: $\frac{c_n}{\sqrt{\mathbb{E}(|c_n|^{2})}} \xrightarrow{d} \sqrt{\mathcal{M}_{\theta}}\,\mathcal{Z}$, with $\mathcal{Z}$ independent of $\mathcal{M}_{\theta}$. The authors extend this to process-level convergence for polynomial functionals $X_n[p]$ and to joint limits of consecutive coefficients governed by a Toeplitz covariance $\mathcal{H}$ derived from the GMC. The approach combines a refined martingale approximation, martingale central limit theorems, and Ewens sampling formula machinery, anchoring fluctuations to the GMC mass while the Gaussian driver remains independent. As a corollary, convergence in law of sublinear-index secular coefficients for the circular-$\beta$-ensemble is obtained for all $\beta>2$, linking random matrix secular data to HMC limits across the subcritical regime.

Abstract

The holomorphic multiplicative chaos (HMC) is a holomorphic analogue of the Gaussian multiplicative chaos. It arises naturally as the limit in large matrix size of the characteristic polynomial of Haar unitary, and more generally circular-$β$-ensemble, random matrices. We consider the Fourier coefficients of the holomorphic multiplicative chaos in the $L^1$-phase, and we show that appropriately normalized, this converges in distribution to a complex normal random variable, scaled by the total mass of the Gaussian multiplicative chaos measure on the unit circle. We further generalize this to a process convergence, showing the joint convergence of consecutive Fourier coefficients. As a corollary, we derive convergence in law of the secular coefficients of sublinear index of the circular-$β$-ensemble for all $β> 2$.

The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency

TL;DR

This work analyzes the Fourier coefficients of holomorphic multiplicative chaos (HMC) in the -phase (), showing that normalized coefficients converge in law to a complex Gaussian scaled by the GMC mass: , with independent of . The authors extend this to process-level convergence for polynomial functionals and to joint limits of consecutive coefficients governed by a Toeplitz covariance derived from the GMC. The approach combines a refined martingale approximation, martingale central limit theorems, and Ewens sampling formula machinery, anchoring fluctuations to the GMC mass while the Gaussian driver remains independent. As a corollary, convergence in law of sublinear-index secular coefficients for the circular--ensemble is obtained for all , linking random matrix secular data to HMC limits across the subcritical regime.

Abstract

The holomorphic multiplicative chaos (HMC) is a holomorphic analogue of the Gaussian multiplicative chaos. It arises naturally as the limit in large matrix size of the characteristic polynomial of Haar unitary, and more generally circular--ensemble, random matrices. We consider the Fourier coefficients of the holomorphic multiplicative chaos in the -phase, and we show that appropriately normalized, this converges in distribution to a complex normal random variable, scaled by the total mass of the Gaussian multiplicative chaos measure on the unit circle. We further generalize this to a process convergence, showing the joint convergence of consecutive Fourier coefficients. As a corollary, we derive convergence in law of the secular coefficients of sublinear index of the circular--ensemble for all .

Paper Structure

This paper contains 9 sections, 12 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Process convergence
  3. The circular $\beta$-ensemble
  4. Preliminaries from najnudel2023secular and the Ewens sampling formula
  5. Martingale approximation and convergence
  6. Martingale approximation
  7. Martingale convergence
  8. Proof of main theorem using the martingale CLT
  9. Joint convergence of the shifted coefficients

Key Result

Theorem 1.1

For any $0 < \theta < 1$, we have the convergence in distribution where $\mathcal{Z}$ and $\mathcal{M}_{\theta}$ are independent, $\mathcal{Z}$ is standard complex normal, and $\mathcal{M}_{\theta}$ has law eq:mass.

Theorems & Definitions (20)

  • Theorem 1.1: $L^{1}$-phase
  • Theorem 1.2: $L^{1}$-phase-multipoint
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1: Martingale approximation
  • ...and 10 more