Invariance principle for the Gaussian Multiplicative Chaos via a high dimensional CLT with low rank increments

TL;DR

This paper proves a universality principle for Gaussian multiplicative chaos by establishing a high-dimensional CLT for low-rank increments, showing non-Gaussian models are mutually absolutely continuous with Gaussian ones.

Contribution

It introduces a new high-dimensional CLT for sums of low-rank dependent vectors and applies it to demonstrate the universality of GMC measures in non-Gaussian settings.

Findings

Gaussian and non-Gaussian chaos measures are mutually absolutely continuous in the sub-critical regime.

A new high-dimensional CLT with error bounds for low-rank sums is developed.

The proof involves novel path-wise analysis and matrix perturbation techniques.

Abstract

Gaussian multiplicative chaos (GMC) is a canonical random fractal measure obtained by exponentiating log-correlated Gaussian processes, first constructed in the seminal work of Kahane (1985). Since then it has served as an important building block in constructions of quantum field theories and Liouville quantum gravity. However, in many natural settings, non-Gaussian log-correlated processes arise. In this paper, we investigate the universality of GMC through an invariance principle. We consider the model of a random Fourier series, a process known to be log-correlated. While the Gaussian Fourier series has been a classical object of study, recently, the non-Gaussian counterpart was investigated and the associated multiplicative chaos constructed by Junnila in 2016. We show that the Gaussian and non-Gaussian variables can be coupled so that the associated chaos measures are almost surely mutually absolutely continuous throughout the entire sub-critical regime. This solves the main open problem from Kim and Kriechbaum (2024) who had earlier established such a result for a part of the regime. The main ingredient is a new high dimensional CLT for a sum of independent (but not i.i.d.) random vectors belonging to rank one subspaces with error bounds involving the isotropic properties of the covariance matrix of the sum, which we expect will find other applications. The proof relies on a path-wise analysis of Skorokhod embeddings as well as a perturbative result about square roots of positive semi-definite matrices which, surprisingly, appears to be new.