The curvature effect in Gaussian random fields

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, mean and principal curvatures of the parametric space, analyzing how these quantities change along dynamics exhibiting phase transitions. During the simulations, we have observed an unexpected phenomenon that we called the 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. When the system undergoes a phase transition from randomness to clustered behavior the curvature is smaller than that observed in the reverse phase transition. This asymmetric pattern relates to the emergence of hysteresis phenomenon, leading to an intrinsic arrow of time along the random field dynamics.