H\"older type estimates for Gaussian multiplicative chaos

TL;DR

This paper studies the tail behavior of Gaussian multiplicative chaos ratios, establishing bounds and optimal exponents, which advances understanding of their probabilistic properties beyond existing inequalities.

Contribution

It introduces a new approach to analyze GMC ratios' tail behavior, overcoming limitations of Kahane's inequality, and provides bounds for their right tail distribution.

Findings

Derived upper and lower bounds for GMC ratios' right tail

Identified the optimal tail exponent for GMC ratios

Extended the class of GMC ratios for analysis

Abstract

We investigate the right tail behavior of a certain class of GMC ratios, reminiscent of H\"older's inequality. We start with a heuristic argument to justify the optimal exponent in the tail estimate. Since Kahane's convexity inequality does not apply to GMC ratios, implementing the heuristic in the continuous setting is nontrivial from the viewpoint of GMC theory. We address the problem by enlarging the class of GMC ratios considered, and deduce the upper and lower bounds for the right tail of GMC ratios.