Stability of cellular automata trajectories revisited: branching walks and Lyapunov profiles

TL;DR

This paper investigates the instability of cellular automaton trajectories by analyzing defect accumulation dynamics through Lyapunov profiles, introducing new computational methods and connecting to large deviation theory.

Contribution

It introduces a variational approach to compute Lyapunov profiles for cellular automata, extending Floquet theory to this context.

Findings

Lyapunov profiles vary exponentially in different directions.

New variational methods effectively compute profiles for periodic states.

Profiles relate to large deviation principles in cellular automata.

Abstract

We study non-equilibrium defect accumulation dynamics on a cellular automaton trajectory: a branching walk process in which a defect creates a successor on any neighborhood site whose update it affects. On an infinite lattice, defects accumulate at different exponential rates in different directions, giving rise to the Lyapunov profile. This profile quantifies instability of a cellular automaton evolution and is connected to the theory of large deviations. We rigorously and empirically study Lyapunov profiles generated from random initial states. We also introduce explicit and computationally feasible variational methods to compute the Lyapunov profiles for periodic configurations, thus developing an analogue of Floquet theory for cellular automata.