A family of probability distributions consistent with the DOZZ formula: towards a conjecture for the law of 2D GMC

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