Bulk/boundary quotients of Gaussian multiplicative chaos measures II: Tail profile of bulk Gaussian multiplicative chaos measures in the exact scale-invariant case

Abstract

This is the second part of a series of papers where we consider questions related to the tail profile of the bulk/boundary quotients of Gaussian multiplicative chaos measures appearing in boundary Liouville conformal field theory. In this part, we study of the right tail profile of a Gaussian multiplicative chaos measure with uniform singularity on the boundary, especially in the case of an exact scale-invariant kernel for the underlying log-correlated Gaussian field. This extends previous results by Rhodes-Vargas and Wong, where the case with flat background geometry (i.e. no boundary singularity) is studied using either the localization trick or some suitable versions of Tauberian theorems. Our generalization is non-trivial in the sense that we don't apply the localization trick to the bulk measure in question, but rather to an auxiliary Gaussian multiplicative chaos measure located at the boundary, on which the original bulk measure is not directly defined. We show that this modified localization scheme correctly captures the behavior of the right tail of the bulk Gaussian multiplicative chaos measure. The resulting tail profile coefficient is expressed in terms of a variant of bulk/boundary quotient of respective Gaussian multiplicative chaos measures, for which the well-definedness follows from preliminary joint moment bounds established previously in a companion paper.