Tail Profile of Bulk Gaussian Multiplicative Chaos Measures I: Bulk/Boundary Quotients

TL;DR

This paper investigates the tail behavior of bulk Gaussian multiplicative chaos measures with boundary singularities, establishing joint moment bounds and generalizing key inequalities to set the stage for future detailed tail profile analysis.

Contribution

It introduces a localization technique at the boundary and extends Kahane's convexity inequality, providing foundational tools for analyzing tail profiles in boundary Liouville conformal field theory.

Findings

Established joint moment bounds for bulk/boundary quotients

Developed a boundary localization trick for Gaussian chaos measures

Generalized Kahane's convexity inequality for boundary measures

Abstract

This is the first part of a series of papers devoted to studying the right tail profile of a bulk Gaussian multiplicative chaos measure with uniform singularity on the boundary. We investigate the bulk/boundary quotients of Gaussian multiplicative chaos measures appearing in boundary Liouville conformal field theory, for which we establish preliminary joint moment bounds. These moment bounds will be a crucial ingredient in establishing the right tail profile of the bulk Gaussian multiplicative chaos measure in subsequent papers. The main idea is to implement the so-called localization trick at the boundary, and we also record a useful generalization of Kahane's convexity inequality, which is of independent interest. The study of the universal tail profiles of general bulk measures and bulk/boundary quotients as well as connections to integrability results of boundary Liouville conformal field theory will be pursued in subsequent papers.