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

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.