# Information geometry, simulation and complexity in Gaussian random fields

## TL;DR

This paper explores how the geometry of Gaussian random fields evolves with system entropy changes, revealing irreversibility and hysteresis-like behavior in the metric tensor during state transitions.

## Contribution

It introduces Fisher curves to quantify geometric deformations in the parametric space of Gaussian random fields, linking entropy variation to intrinsic irreversibility.

## Key findings

- Decreases in inverse temperature cause irreversible geometric deformations.
- Fisher curves exhibit hysteresis-like behavior during entropy cycles.
- System evolution has a natural arrow of time due to geometric irreversibility.

## Abstract

Random fields are useful mathematical objects in the characterization of non-deterministic complex systems. A fundamental issue in the evolution of dynamical systems is how intrinsic properties of such structures change in time. In this paper, we propose to quantify how changes in the spatial dependence structure affect the Riemannian metric tensor that equips the model's parametric space. Defining Fisher curves, we measure the variations in each component of the metric tensor when visiting different entropic states of the system. Simulations show that the geometric deformations induced by the metric tensor in case of a decrease in the inverse temperature are not reversible for an increase of the same amount, provided there is significant variation in the system entropy: the process of taking a system from a lower entropy state A to a higher entropy state B and then bringing it back to A, induces a natural intrinsic one-way direction of evolution. In this context, Fisher curves resemble mathematical models of hysteresis in which the natural orientation is pointed by an arrow of time.