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

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.

Figures (9)

• Figure 1: Structure of the 1+1D classical circuit. Each dot represents the value $s_i(t)$ of a spin at a certain time. Arrows show which neighbor is involved in the update of a given spin at a given time, Eqs. \ref{['eq1plus1update']}, \ref{['eq1plus1update2']}.
• Figure 2: Hasse diagrams for $\Pi_3$ and $\Pi_4$stanley_enumerative_1999: ${\pi< \sigma}$ iff $\pi$ can be reached from $\sigma$ by following the edges of the graph upward. The node at the bottom of a given diagram represents the partition ${\mathbbm{1}\in \Pi_n}$, i.e. an undamaged state where all replicas agree. The node at the top represents the partition $(1)(2)\ldots(n)$, i.e. a state where every replica differs from every other.
• Figure 3: Top left: a damage trajectory for three replicas with the three nontrivial partitions of the form ${(a)(bc)}$ represented by three colors (Eq. \ref{['eq:n3states']}). Other panels: neglecting any one of the three replicas gives a two-replica damage trajectory, as explained below Eq. \ref{['eq:3siteupdatepictures']}. Here $L=256$ and $t=256$.
• Figure 4: The density of damages, $\varrho_{n}(t)$, for $n=2$, $3$, and $4$ replicas, as a function of time $t$ at the critical point. Different colours correspond to the different numbers of replicas, $n$, whereas different system sizes, $L$, are denoted by different markers (and colour intensities). The dashed lines show the fits used to extract the $\alpha_n$ exponents, values of which are also mentioned in the figures. All data are averaged over 500 realisations of the dynamics.
• Figure 5: The survival probability, ${\cal S}_n(t)$, defined in Eq. \ref{['eq:surv-prob']}, as a function of time $t$, for different numbers of replicas $n$ (different colours), and different system sizes $L$ (different markers and colour intensities) at the critical point. The dashed lines denote fits to extract the $\delta_n$ exponent as ${\cal S}_n(t)\sim t^{-\delta_n}$ with the values mentioned in the figure. All data are averaged over 1000 realisations of the dynamics.