Gaussian multiplicative chaos and KPZ duality

TL;DR

This paper constructs atomic Gaussian multiplicative chaos measures beyond the critical phase and proves duality relations, providing new insights into Liouville quantum gravity and simplifying the KPZ formula.

Contribution

It introduces a method to construct atomic Gaussian multiplicative chaos measures for super-critical parameters and establishes duality relations with sub-critical measures.

Findings

Constructed purely atomic measures for b3^2 > 2d

Proved duality relations between atomic and sub-critical chaos

Provided simplified proofs of KPZ and dual KPZ formulas

Abstract

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter $\gamma^2$ beyond the transition phase (i.e. $\gamma^2>2d$) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity.