Peierls bounds from Toom contours

TL;DR

The paper develops a streamlined Peierls-type argument using Toom contours to study stability of monotone cellular automata under intrinsic randomness, and extends the approach to derive explicit lower bounds on the critical noise parameter for deterministic rules. It introduces a general framework with edge speeds and polar functions, constructs Toom contours and cycles, and proves that the presence of a contour bounds the probability that the maximal trajectory at the origin remains zero. The results yield concrete stability conditions and quantitative bounds for several CA rules (notably coop and NEC), and connect the contour method to bootstrap percolation phenomena. Collectively, the work provides a tractable, contour-based method for assessing robustness of monotone CA against noise and offers a path toward sharper bounds in a companion paper.

Abstract

For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. The proof of sufficiency is based on an intricate Peierls argument. We present a simplified version of this Peierls argument. Our main motivation is the open problem of determining stability of monotone cellular automata with intrinsic randomness, in which for the unperturbed evolution the local update rules at different space-time points are chosen in an i.i.d. fashion according to some fixed law. We apply Toom's Peierls argument to prove stability of a class of cellular automata with intrinsic randomness and also derive lower bounds on the critical parameter for some deterministic cellular automata.

Figures (5)

• Figure 1: Density $\overline\rho$ of the upper invariant law of two monotone random cellular automata as a function of the parameters, shown on a scale from 0 (white) to 1 (black). On the left: a version of Toom's model that applies the maps $^0$, $^1$, and $^{\rm NEC}$ with probabilities $p$, $q$, and $1-p-q$, respectively. On the right: the mononotone random cellular automaton that applies the maps $^0$, $^1$, and $^{\rm NN}$ with probabilities $p$, $q$, and $1-p-q$, respectively. The map $^{\rm NEC}$ is an eroder but $^{\rm NN}$ is not. By the symmetry between the 0's and the 1's, in both models, the density $\underline\rho(p, q)$ of the lower invariant law equals $1-\overline\rho(q,p)$. Due to metastability effects, the area where the upper invariant law differs from the lower invariant law is shown too large in these numerical data. For Toom's model with $q=0$, the data shown above suggest a first order phase transition at $p_{\rm c}\approx 0.057$ but based on numerical data for edge speeds we believe the true value is $p_{\rm c}\approx 0.053$. We conjecture that the model on the right has a unique invariant law everywhere except on the diagonal $p=q$ for $p$ sufficiently small.
• 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: A Toom contour in ${\mathbb Z}^3$ rooted at $(0,0,0)$. The third coordinate represents time and is plotted downwards. The picture on the right shows a minimal explanation (or rather its associated undirected explanation graph as defined in Subsection \ref{['S:minexpl']}) for a monotone cellular automaton $\Phi^p$ that applies the maps $^0$ and $^{\rm coop}$ with probabilities $p$ and $1-p$, respectively. The origin has the value zero because the sites marked with a star are defective; removing any of these defective sites results in the origin having the value one. The Toom contour in the middle picture is present in $\Phi^p$. In particular, the sinks of the Toom contour coincide with some, though not with all of the defective sites of the minimal explanation.
• Figure 4: Embedding of a rooted Toom graph inside a typed explanation graph. On the right: a typed explanation graph $(U,{\cal G})$ associated with a minimal explanation for $(0,0,0)$ in the sense of Proposition \ref{['P:typexpl']}. On the left and in the middle: embedding of a rooted Toom graph in $(U,{\cal G})$ in the sense of Theorem \ref{['T:embed']}. The connected component of this Toom graph containing the root is a Toom contour rooted at $(0,0,0)$ (compare Figure \ref{['fig:minexpl1']}).
• Figure 5: The process of exploration and loop erasion. The Toom cycle is constructed on the explanation graph of Figure \ref{['fig:minexpl2']}. We can see that in the Toom cycle on the left $v$ is a sink, but $\psi_v=i$ is not a defective site. In the exploration step, $v$ is replaced by two internal vertices, one of each charge, and two new sinks are added to the cycle at the positions $j_1$ and $j_2$. This leads to the new sink at $j_2$ overlapping with a preexisting sink. In the loop erasion step, this is resolved by erasing the part of the cycle between the first and second visit to $j_2$.

Theorems & Definitions (47)

• Definition 1: Eroders
• Theorem 2: Toom's stability theorem
• Proposition 3: Erosion criterion
• Remark 4
• Remark 5
• Definition 6: Edge speed
• Lemma 7: Edge speeds
• Definition 8: Polar functions
• Theorem 9: Stability of monotone cellular automata with intrinsic randomness
• Lemma 10: Alternative erosion criterion