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

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 $(2,2).$

27 pages. To appear in J. Stat. Phys

References in corpus (3)

• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Freezing and decorated Poisson point processes
• High values of disorder-generated multifractals and logarithmically correlated processes

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