Squeezing codes: robust fluctuation-stabilized memories

TL;DR

This work introduces squeezing codes, simple local probabilistic cellular automata that autonomously stabilize order and memory in $d\ge2$, achieving robustness via fluctuation-stabilized dynamics distinct from Toom-like rules. By combining dynamic mean-field, cluster variational methods, master-equation formalisms, and Floquet-Glauber realizations, the authors show that memory phases can persist under asynchronous noise for several codes, with rich non-equilibrium critical behavior and domain-wall dynamics. They demonstrate that order can be enhanced by fluctuations (counter to naive MF expectations), identify new non-equilibrium universality classes with $z<2$, and uncover synchronicity-protected regimes where memory persists only above a critical synchronization threshold. The results illuminate mechanisms for robust memories and propose a framework for classifying noise-robust dynamical phases, with potential relevance to non-equilibrium statistical mechanics and active-matter analogies.

Abstract

We introduce families of classical stochastic dynamics in two and higher dimensions which stabilize order in the absence of any symmetry. Our dynamics are qualitatively distinct from Toom's rule, and have the unusual feature of being fluctuation-stabilized: their order becomes increasingly fragile in larger dimensions. One of our models maintains an ordered phase only in two dimensions. The phase transitions that occur as the order is lost realize new dynamical universality classes which are fundamentally non-equilibrium in character.

Figures (18)

• Figure 1: a) Error correction in the $\mathsf{R}$ squeezing code, shown for noiseless asynchronous updates. The top panels: a $+1$ (white) minority domain is annihilated by being squeezed in the direction indicated by the arrows. The bottom panels show the squeezing experienced by a $-1$ (black) minority domain, which is related to the process in the top panels by the combination of a spin flip and a $\pi/2$ rotation. b) the schematic phase diagram of squeezing codes as a function of noise strength $p$ and bias $\eta$. The phase transition out of the memory phase is first order at $\eta \neq 0$ and second order at $\eta = 0$. The pink star at zero bias denotes a new type of non-equilibrium critical point, which is discussed in Sec. \ref{['sec:crit']}.
• Figure 2: Schematic definitions of the squeezing codes studied in this work. In each panel, the colored site (square) is updated to be the $\wedge$ (${\tt and}$) or the $\vee$ (${\tt or}$) of the indicated squares, as appropriate.
• Figure 3: The erosion of minority domains of $1$s in systems of size $L = 350$ under the different types of squeezing dynamics studied in this paper. Time runs left to right, and different time scales are used in each row. The top rows in each panel show synchronous updates, and the bottom rows show asynchronous updates. The rule $\mathsf{T}$ fails to erode the minority domain under asynchronous updates, with the system rapidly "self-destructing".
• Figure 4: Linear erosion under asynchronous updates in the automata $\mathsf{R},\mathsf{F},\mathsf{M}$. $t_{\sf erode}(R)$ is the average time taken by noiseless asynchronous dynamics to erode a circular domain of size $R$. Each data point is based on 1000 samples. The dashed lines are linear fits.
• Figure 5: Phase transitions in the memory time under asynchronous $\mathsf{R}$ dynamics, at zero bias (left) and maximal bias (right). Error bars are smaller than the data points.

Theorems & Definitions (9)

• Definition 1: eroders
• Definition 2: memory time
• Theorem 1
• Theorem 2
• Proposition 1: unique absorbing states
• proof
• proof
• proof
• proof