On the multiplicative chaos of non-Gaussian log-correlated fields

TL;DR

This paper extends the theory of multiplicative chaos to non-Gaussian log-correlated fields, establishing convergence, moments, and analyticity in the subcritical regime, with applications to non-Gaussian Fourier series.

Contribution

It generalizes Gaussian multiplicative chaos results to non-Gaussian fields satisfying specific moment and regularity conditions.

Findings

Proves convergence and existence of moments for non-Gaussian chaos.

Establishes analyticity with respect to inverse temperature.

Applies results to non-Gaussian Fourier series.

Abstract

We study non-Gaussian log-correlated multiplicative chaos, where the random field is defined as a sum of independent fields that satisfy suitable moment and regularity conditions. The convergence, existence of moments and analyticity with respect to the inverse temperature are proven for the resulting chaos in the full subcritical range. These results are generalizations of the corresponding theorems for Gaussian multiplicative chaos. A basic example where our results apply is the non-Gaussian Fourier series \[\sum_{k=1}^\infty \frac{1}{\sqrt{k}}(A_k \cos(2\pi k x) + B_k \sin(2\pi k x)),\] where $A_k$ and $B_k$ are i.i.d. random variables.