Moment bounds for Gaussian multiplicative chaos with higher-dimensional singularities

TL;DR

This paper establishes the precise threshold for the existence of moments in Gaussian multiplicative chaos with higher-dimensional singularities, advancing understanding in boundary Liouville conformal field theory and related probabilistic models.

Contribution

It provides the exact moment threshold for Gaussian multiplicative chaos with arbitrary singularities, improving previous bounds and confirming conjectured optimality.

Findings

Determined the threshold for positive moments with higher-dimensional singularities.

Improved previous bounds for 0-dimensional singularities.

Proved the optimality of the moment bound threshold for boundary chaos.

Abstract

We determine the exact threshold of the extended Seiberg bound for the existence of correlation functions in the boundary Liouville conformal field theory in the unit disk. In probabilistic terms, our result is a toolbox yielding the threshold for the existence of positive moments of Gaussian multiplicative chaos measure, appliable to the case where singularities of arbitrary (co-)dimension in the background metric are present. We improve previous results of this type for 0-dimensional singularities in [DKRV16] and a sufficient condition for the 1-dimensional singularity in an unpublished appendix of [HRV18]. In particular, we prove the optimality of the moment bound threshold for boundary Gaussian multiplicative chaos conjectured in [HRV18], which is equivalent to the so-called unit volume Seiberg bound of the boundary Liouville conformal field theory.