Butterfly Effect and Spatial Structure of Information Spreading in a Chaotic Cellular Automaton

Abstract

Inspired by recent developments in the study of chaos in many-body systems, we construct a measure of local information spreading for a stochastic Cellular Automaton in the form of a spatiotemporally resolved Hamming distance. This decorrelator is a classical version of an Out-of-Time-Order Correlator studied in the context of quantum many-body systems. Focusing on the one-dimensional Kauffman Cellular Automaton, we extract the scaling form of our decorrelator with an associated butterfly velocity $v_b$ and a velocity-dependent Lyapunov exponent $\lambda (v)$. The existence of the latter is not a given in a discrete classical system. Second, we account for the behaviour of the decorrelator in a framework based solely on the boundary of the information spreading, including an effective boundary random walk model yielding the full functional form of the decorrelator. In particular, we obtain analytic results for $v_b$ and the exponent $\beta$ in the scaling ansatz $\lambda (v)\sim \mu (v-v_b{)}^{\beta }$, which is usually only obtained numerically. Finally, a full scaling collapse establishes the decorrelator as a unifying diagnostic of information spreading.

arXiv comment5+1 pages, 4 figures

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• Constrained Dynamics and Directed Percolation
• Quantum and Classical Dynamics with Random Permutation Circuits
• Absence of localization in two-dimensional Clifford circuits
• Classical route to ergodicity and scarring in collective quantum systems
• Semiclassical Limit of Measurement-Induced Transition in Many-Body Chaos in Integrable and Nonintegrable Oscillator Chains
• Random Permutation Circuits Beyond Qubits are Quantum Chaotic
• Localization of information driven by stochastic resetting
• Phase transition from localization to chaos in classical many-body system