An elementary approach to Gaussian multiplicative chaos

TL;DR

This paper provides an elementary, self-contained proof of Gaussian multiplicative chaos convergence, demonstrating the nontriviality and universality of the limiting measure across the entire subcritical phase.

Contribution

It introduces a simple, elementary proof technique for Gaussian multiplicative chaos convergence, emphasizing universality and nontriviality in the subcritical phase.

Findings

Proves convergence of Gaussian multiplicative chaos using elementary methods

Shows the limiting measure is nontrivial for all b3 < 2d

Establishes the universality of the limit regardless of regularization

Abstract

A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase $(\gamma < \sqrt{2d})$ and that the limit is universal (i.e., the limiting measure is independent of the regularisation of the underlying field)