On the Fourier coefficients of critical Gaussian multiplicative chaos

TL;DR

This paper investigates the asymptotic behavior of Fourier coefficients of critical Gaussian multiplicative chaos, showing they decay at a specific rate involving a logarithmic factor.

Contribution

It extends the understanding of Fourier coefficients in critical GMC by establishing their decay rate and convergence properties.

Findings

$( ext{log } n)^{eta} c_n$ converges to zero in probability for $eta<1/4$

Provides new insights into the structure of critical Gaussian multiplicative chaos

Advances theoretical understanding of Fourier analysis in stochastic processes

Abstract

We continue the study of the Fourier coefficients of Gaussian multiplicative chaos (GMC) recently initiated by Garban and Vargas. We show that if $\{c_n\}_{n\geq 1}$ are the Fourier coefficients of critical GMC on the unit interval, then $(\log n)^{\alpha}c_n$ converges to zero in probability as $n$ tends to infinity for any $\alpha<1/4$.