A Family of Probability Distributions Consistent with the DOZZ Formula: Towards a Conjecture for the Law of 2D GMC

Abstract

A three parameter family of probability distributions is constructed such that its Mellin transform is defined over the same domain as the 2D GMC on the Riemann sphere with three insertion points $(\alpha_1,\alpha_2,\alpha_3)$ and satisfies the DOZZ formula in the sense of Kupiainen (Ann. Math. 191 (2020) 81 -- 166). The probability distributions in the family are defined as products of independent Fyodorov-Bouchaud and powers of Barnes beta distributions of types $(2, 1)$ and $(2, 2).$ In the special case of $\alpha_1+\alpha_2+\alpha_3=2Q$ the constructed probability distribution is shown to be consistent with the known small deviation asymptotic of the 2D GMC laws with everywhere positive curvature.