A Review of Conjectured Laws of Total Mass of Bacry-Muzy GMC Measures on the Interval and Circle and Their Applications

TL;DR

This paper reviews conjectured laws related to the total mass of Bacry-Muzy Gaussian Multiplicative Chaos measures on the interval and circle, exploring their properties and applications in probability and mathematical physics.

Contribution

It provides a comprehensive review of the analytic, asymptotic, and probabilistic properties of Selberg and Morris integral distributions, including new proofs and applications.

Findings

Detailed analysis of Mellin transform representations

Proofs of infinite divisibility and factorizations into Barnes beta distributions

Applications to limit theorems, martingales, and noise extrema

Abstract

Selberg and Morris integral probability distributions are long conjectured to be distributions of the total mass of the Bacry-Muzy Gaussian Multiplicative Chaos measures with non-random logarithmic potentials on the unit interval and circle, respectively. The construction and properties of these distributions are reviewed from three perspectives: analytic based on several representations of the Mellin transform, asymptotic based on low intermittency expansions, and probabilistic based on the theory of Barnes beta probability distributions. In particular, positive and negative integer moments, infinite factorizations and involution invariance of the Mellin transform, analytic and probabilistic proofs of infinite divisibility of the logarithm, factorizations into products of Barnes beta distributions, and Stieltjes moment problems of these distributions are presented in detail. Applications are given in the form of conjectured mod-Gaussian limit theorems, laws of derivative martingales, distribution of extrema of $1/f$ noises, and calculations of inverse participation ratios in the Fyodorov-Bouchaud model.