# Universal tail profile of Gaussian multiplicative chaos

## TL;DR

This paper establishes a universal asymptotic description of the tail probability for Gaussian multiplicative chaos across various dimensions, resolving a conjecture and introducing a novel analytical approach.

## Contribution

It introduces a unified method using Tauberian and renewal theorems to derive precise tail asymptotics for Gaussian multiplicative chaos in any dimension.

## Key findings

- Derived first order tail asymptotics for Gaussian multiplicative chaos
- Identified a universal constant governing tail behavior
- Resolved a conjecture by Rhodes and Vargas

## Abstract

In this article we study the tail probability of the mass of Gaussian multiplicative chaos. With the novel use of a Tauberian argument and Goldie's implicit renewal theorem, we provide a unified approach to general log-correlated Gaussian fields in arbitrary dimension and derive precise first order asymptotics of the tail probability, resolving a conjecture of Rhodes and Vargas. The leading order is described by a universal constant that captures the generic property of Gaussian multiplicative chaos, and may be seen as the analogue of the Liouville unit volume reflection coefficients in higher dimensions.