On Barnes Beta Distributions and Applications to the Maximum Distribution of the 2D Gaussian Free Field

TL;DR

This paper introduces a new family of Barnes beta distributions, explores their mathematical properties, and connects them to the maximum distribution of the 2D Gaussian free field, proposing conjectural links to Selberg and Morris integrals.

Contribution

It develops a novel family of Barnes beta distributions and applies them to describe the maximum distribution of the 2D Gaussian free field, establishing new mathematical properties and conjectural relationships.

Findings

New Barnes beta distributions with established properties

Construction of Morris integral probability distribution from Barnes beta distributions

Conjectural relationships between maximum distributions and integral probability distributions

Abstract

A new family of Barnes beta distributions on $(0, \infty)$ is introduced and its infinite divisibility, moment determinacy, scaling, and factorization properties are established. The Morris integral probability distribution is constructed from Barnes beta distributions of types $(1,0)$ and $(2,2),$ and its moment determinacy and involution invariance properties are established. For application, the maximum distributions of the 2D gaussian free field on the unit interval and circle with a non-random logarithmic potential are conjecturally related to the critical Selberg and Morris integral probability distributions, respectively, and expressed in terms of sums of Barnes beta distributions of types $(1,0)$ and $(2,2).$