The Curvature Effect in Gaussian Random Fields

TL;DR

This paper investigates the geometric properties of Gaussian random fields, revealing a curvature effect linked to phase transitions and hysteresis, which provides insights into the system's intrinsic arrow of time.

Contribution

It introduces the concept of the curvature effect in Gaussian random fields and analyzes its relation to phase transitions and hysteresis using geometric and simulation methods.

Findings

Identification of the curvature effect during phase transitions

Observation of asymmetric geometric deformation in parameter space

Link between curvature changes and the emergence of hysteresis

Abstract

Random field models are mathematical structures used in the study of stochastic complex systems. In this paper, we compute the shape operator of Gaussian random field manifolds using the first and second fundamental forms (Fisher information matrices). Using Markov Chain Monte Carlo techniques, we simulate the dynamics of these random fields and compute the Gaussian curvature of the parametric space, analyzing how this quantity changes along phase transitions. During the simulation, we have observed an unexpected phenomenon that we called the \emph{curvature effect}, which indicates that a highly asymmetric geometric deformation happens in the underlying parametric space when there are significant increase/decrease in the system's entropy. This asymmetric pattern relates to the emergence of hysteresis, leading to an intrinsic arrow of time along the dynamics.