On the curvatures of Gaussian random field manifolds

TL;DR

This paper derives explicit formulas for the curvature of Gaussian random field manifolds and links curvature changes to phase transitions and irreversible dynamics, revealing intrinsic temporal properties.

Contribution

It provides the first closed-form expressions for fundamental forms and curvatures of Gaussian-Markov random field manifolds, connecting geometry with phase transitions.

Findings

Curvature sign change correlates with phase transitions.

Curvatures exhibit asymmetry for positive and negative inverse temperature displacements.

Curvature dynamics suggest an intrinsic notion of time in the field evolution.

Abstract

Information geometry is concerned with the application of differential geometry concepts in the study of the parametric spaces of statistical models. When the random variables are independent and identically distributed, the underlying parametric space exhibit constant curvature, which makes the geometry hyperbolic (negative) or spherical (positive). In this paper, we derive closed-form expressions for the components of the first and second fundamental forms regarding pairwise isotropic Gaussian-Markov random field manifolds, allowing the computation of the Gaussian, mean and principal curvatures. Computational simulations using Markov Chain Monte Carlo dynamics indicate that a change in the sign of the Gaussian curvature is related to the emergence of phase transitions in the field. Moreover, the curvatures are highly asymmetrical for positive and negative displacements in the inverse temperature parameter, suggesting the existence of irreversible geometric properties in the parametric space along the dynamics. Furthermore, these asymmetric changes in the curvature of the space induces an intrinsic notion of time in the evolution of the random field.