Peierls bounds from random Toom contours

TL;DR

This work develops a random contour framework to extend Toom’s stability results to monotone cellular automata with intrinsic randomness on countable groups. By introducing random Toom contours and decorated contours, the authors obtain abstract bounds that bound the Peierls sum and yield explicit stability results for several nontrivial models, including NEC Toom rule, triangular-lattice majority, and cooperative branching with an identity map. The method also exposes limits of the approach through constructions where the Peierls sum diverges, highlighting annealed vs quenched aspects and the necessity of further refinements. Overall, the paper provides both concrete stability criteria and a deeper understanding of the contour-based mechanisms controlling robustness against random perturbations in random CA dynamics.

Abstract

For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom (1980) has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. We are interested in the open problem of extending Toom's result to monotone cellular automata with intrinsic randomness, where the unperturbed evolution is random with i.i.d. update rules attached to the space-time points. For some applications it is also desirable to consider a more general graph structure, so we assume that the underlying lattice is an arbitrary countable group. Toom's proof of stability is based on a Peierls argument. In previous work, we demonstrated that this Peierls argument can also be used to prove stability for cellular automata with intrinsic randomness, but in this case estimating the Peierls sum becomes much harder than in the deterministic case. In the present paper, we develop a method based on random contours to estimate the Peierls sum and apply it to prove new stability results for monotone cellular automata with intrisic randomness. We also demonstrate the limitations of the method by constructing an example where the Peierls sum is infinite for arbitrary small perturbations even though stability is believed to hold.

Figures (4)

• Figure 1: A monotone cellular automaton with intrinsic randomness on the triangular lattice. The value of the point in the middle is replaced by the value that holds the majority in a set chosen uniformly at random from the three sets drawn in the picture.
• Figure 2: Example of a Toom graph with three charges. Sources are indicated with open dots, sinks with asterixes, and internal vertices and edges of the three possible charges with three colours. Note the isolated vertex in the lower right corner, which is a source and a sink at the same time.
• Figure 3: The left and middle picture show a Toom contour in ${\mathbb Z}^3$ rooted at $(0,0,0)$, that is present in the typed dependence graph $(\Lambda,{\cal H})$ defined in (\ref{['Hicoop']}). The associated monotone cellular automaton $\Phi^p$ applies the constant map $^0$ in the set $\Lambda_0$ of defective space-time points and the map $^{\rm coop}$ from (\ref{['coopid']}) in all other points. In the picture on the right defective sites are marked with a star and the black lines help us understand that the maximal trajectory satisfies $\overline X^p(0,0,0)=0$. Sinks of the Toom contour coincide with defective sites. The height function $h(i_1,i_2,i_3):=-i_3$ is plotted upwards in these pictures.
• Figure 4: An incomplete Toom graph with charges $1=$ green, $2=$ red, and $3=$ blue. To stress the special role of charge 3, the direction of edges with this charge has been reversed in the picture. Loose ends are indicated by small arrows and dead ends by small line segments.

Theorems & Definitions (49)

• Lemma 1: Minimal and maximal trajectories
• Theorem 2: Toom's stability theorem
• Theorem 3: Toom's rule
• Theorem 4: Majority rules on the triangular lattice
• Theorem 5: Cooperative branching and the identity map
• Lemma 6: Edge speeds
• Lemma 7: Erosion criterion
• Theorem 8: Theorem 9 of SST24
• Theorem 9: Theorem 1 of Too80
• Theorem 10: Bound based on a polar function of dimension two