Decoupling and Multipoint moments for the Inverse of the Gaussian multiplicative chaos

TL;DR

This paper investigates the decoupling structure and multipoint moments of the inverse Gaussian multiplicative chaos, aiming to extend conformal welding results to the inverse case and establishing integrability properties of the inverse homeomorphism.

Contribution

It provides new insights into the inverse Gaussian multiplicative chaos, including decoupling structures, multipoint moments, and integrability of the inverse homeomorphism, extending previous conformal welding work.

Findings

Proved the decoupling structure of the inverse chaos.

Established the L^1 integrability of the inverse homeomorphism's dilatation.

Extended conformal welding results to the inverse setting.

Abstract

In this article we study the decoupling structure and multipoint moment of the inverse of the Gaussian multiplicative chaos. It is also the second part of preliminary work for extending the work in "Random conformal weldings" (by K. Astala, P. Jones, A. Kupiainen, E. Saksman) to the existence of Lehto welding for the inverse. In particular, we prove that the dilatation of the inverse homeomorphism on the positive real line is in $L^{1}([0,1]\times[0,2])$.