Characterization of the set of zero-noise limits measures of perturbed cellular automata

TL;DR

The paper characterizes which sets of shift-invariant invariant measures can arise as the zero-noise limits $\mathcal{M}_0^l$ of a cellular automaton perturbed by uniform noise as $\epsilon\to0$. It establishes topological (compactness, connectedness) and computability ($\Pi_3$- and $\Pi_2$-computability) obstructions, and proves a universal realization theorem: every connected $\Pi_2$-computable compact subset $\mathcal{K}$ of invariant measures can be realized as $\mathcal{M}_0^l$ of some CA with uniform noise, with $\mathcal{M}_0^l$ uniformly approached. The construction uses a computation-zone architecture around a star symbol to sequentially approximate the target set by convex combinations of measures supported on periodic orbits, ensuring convergence to $\mathcal{K}$ while maintaining robustness to noise. The work also shows that the zero-noise limit can be highly sensitive to the noise bias, obtaining any connected compact target set by adjusting the bias, and establishes a notion of chaos when $\mathcal{M}_0^l$ is not a singleton. These results bridge dynamical stability, computability theory, and CA design, providing a comprehensive view of how noise shapes observable invariant measures in one-dimensional CA.

Abstract

We add small random perturbations to a cellular automaton and consider the one-parameter family $(F_ε)_{ε>0}$ parameterized by $ε$ where $ε>0$ is the level of noise. The objective of the article is to study the set of limiting invariant distributions as $ε$ tends to zero denoted $\mathcal{M}_0^l$. Some topological obstructions appear, $\mathcal{M}_0^l$ is compact and connected, as well as combinatorial obstructions as the set of cellular automata is countable: $\mathcal{M}_0^l$ is $Π_3$-computable in general and $Π_2$-computable if it is uniformly approached. Reciprocally, for any set of probability measures $\mathcal{K}$ which is compact, connected and $Π_2$-computable, we construct a cellular automaton whose perturbations by an uniform noise admit $\mathcal{K}$ as the zero-noise limits measure and this set is uniformly approached. To finish, we study how the set of limiting invariant measures can depend on a bias in the noise. We construct a cellular automaton which realizes any connected compact set (without computable constraints) if the bias is changed for an arbitrary small value. In some sense this cellular automaton is very unstable with respect to the noise.

Figures (8)

• Figure 3.1: Proof's scheme of Proposition \ref{['prop.MlConnected']}.
• Figure 5.1: Scheme of a space-time diagram the construction (with time going up). Left: a $*$ symbol creates walls that separates into three zones. After $t$ iterations, the central display zone is of linear size, while the other are of logarithmic size. Right: in the display zone, the word $\omega_n$ begins to be sent after $T_n$ iterations. All symbols from $\mathcal{A}$ are shifted along with the left wall.
• Figure 5.2: Ordering of the considered cells ($i>0$ and $-i \leq t < 0$), from left to right and top to bottom. The arrow indicates for each the number of steps $Y$ for a $*$ symbol at this row to reach the grayed cell in position $(0,0)$ in space-time.
• Figure 5.3: The first 16 steps. The $\#$ symbols are omitted after the first step. A trapper signal $T$ is produced on $T_1=4$ and $T_2=16$.
• Figure 5.4: The displaying process. Left to the wall $W$ is the displaying zone, where the symbols are shifted to the left each step. Right to the wall is the computation zone, where the word $\omega = u_1 u_2 u_3 u_4$ is written twice. At each step, the symbols are shifted accordingly so that it forms a loop.

Theorems & Definitions (48)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Definition 3.1
• Lemma 3.2