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

TL;DR

This paper constructs a three-parameter family of probability distributions aligned with the DOZZ formula, providing insights into the law of 2D Gaussian Multiplicative Chaos and its asymptotics on the Riemann sphere.

Contribution

It introduces a novel family of probability distributions consistent with the DOZZ formula, linking GMC laws with Barnes beta distributions and special cases.

Findings

Distributions match the DOZZ formula over the same domain as 2D GMC.

In the case of ++=2Q, distributions align with known small deviation asymptotics.

Constructed distributions are products of Fyodorov-Bouchaud and Barnes beta distributions.

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.