The tail expansion of Gaussian multiplicative chaos and the Liouville reflection coefficient

Abstract

In this short note we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d Gaussian Free Fiedl (GFF) on the unit disk with zero average on the unit circle (and variants). More specifically, we show that to first order the tail is a constant times an inverse power with an explicit value for the tail exponent as well as an explicit value for the constant in front of the inverse power; we also provide a second order bound for the tail expansion. The main interest of our work consists of two points. First, our derivation is based on a simple method, which we believe is universal in the sense that it can be generalized to all dimensions and to all log-correlated fields. Second, in the 2d case we consider, the value of the constant in front of the inverse power is (up to explicit terms) nothing but the Liouville reflection coefficient taken at a special value. The explicit computation of the constant was performed in the recent rigorous derivation with A. Kupiainen of the DOZZ formula (Kupiainen, Rhodes and Vargas (2017a, 2017b)); to our knowledge, it is the first time one derives rigorously an explicit value for such a constant in the tail expansion of a GMC measure. We have deliberately kept this paper short to emphasize the method so that it becomes an easily accessible toolbox for computing tails in GMC theory.