The damage spreading transition: a hierarchy of renormalization group fixed points

TL;DR

This paper develops a comprehensive theory for the damage-spreading transition in cellular automata, revealing a hierarchy of fixed points that extend beyond directed percolation, with implications for nonequilibrium phase transitions.

Contribution

It introduces a hierarchy of renormalization group fixed points for damage spreading, extending the known directed percolation framework to include higher-order observables.

Findings

Hierarchy of fixed points includes directed percolation as a subsector

Higher observables involve overlaps and set partitions

Universal critical exponents characterize the hierarchy

Abstract

Deterministic classical cellular automata can be in two phases, depending on how irreversible the dynamical rules are. In the strongly irreversible phase, trajectories with different initial conditions coalesce quickly, while in the weakly irreversible phase, trajectories with different initial conditions can remain different for a time exponential in the system volume. The transition between these phases is referred to as the damage-spreading transition (the "damaged" sites are those that differ between the trajectories). We develop a theory for this transition. In the simplest and most generic setting, the transition is known to be related to directed percolation, one of the best-studied nonequilibrium phase transitions. However, we show that full theory of the damage-spreading critical point is richer than directed percolation, and contains an infinite hierarchy of sectors of local observables. Directed percolation describes the first level of the hierarchy. The higher observables include "overlaps" for multiple trajectories, and may be labeled by set partitions. (These higher observables arise naturally if, for example, we consider decay of entropy under the irreversible dynamics.) The full hierarchy yields a hierarchy of nonequilibrium fixed points for reaction-diffusion-type processes, all of which contain directed percolation as a subsector, but which possess additional universal critical exponents. We analyze these higher fixed points using a field theory formulation and renormalization group arguments, and using simulations in 1+1 dimensions.