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In this short note, we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d GFF on the unit disk with zero average on the unit circle. 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 in 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 is then made possible thanks to our recent rigorous derivation with Kupiainen of the DOZZ formula \cite{KRV1,KRV}; to our knowledge, it is the first time one gives an explicit value for such a constant in the tail expansion of a GMC measure. The paper is voluntarily very short to emphasize the method so that it becomes a toolbox for computing tails in GMC theory.

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