Renormalization for Robust Memory without Synchronization

Abstract

Local error correction can preserve classical information against noise, but protection under synchronized updates need not survive when sites evolve at independent random times. Establishing stability is particularly challenging when protection relies on the collective action of competing local rules. Here we develop a spacetime renormalization framework that reveals how independently acting local updates can collectively stabilize memory. We identify sufficient conditions under which the geometry of stable interfaces and suppression of defect propagation produce reliable erosion at large scales. Applying this framework, we prove stability of the asynchronous \textsf{R} squeezing code and introduce robust processes in two and three dimensions. In these systems, both encoded phases persist under sufficiently weak noise, even when the noise favors one phase. Our results provide a criterion for designing autonomous memories sustained by competing local stochastic dynamics.

Synchronous local error-correction rules can lose stability when updates occur at independent Poisson times. This paper asks when competing, asynchronous local rules preserve a classical bit under weak noise, including noise biased toward either encoded phase. It treats finite-range, translation-invariant, monotone binary processes without requiring mutually exclusive defect-birth and defect-death rates. The main result is a sufficient stability criterion based on spacetime coarse-graining. A balanced family of invariant clean-half-space directions, with positive net interface advance, defines a deterministic coarse eroder. If adversarial finite-buffer block-failure probabilities vanish with scale and buffers occupy a uniformly bounded number of coarse cells, microscopic evolution is dominated by that eroder with sparse errors. A bounded-dependence argument and Toom’s theorem establish $\lim_{\varepsilon\to0}\sup_{t\ge0}\Pr(\eta_t(x)=1)=0$ from the all-zero state. For the asynchronous two-dimensional $\mathsf R$ squeezing process, comparison of exposed rows with a subcritical one-dimensional contact process proves bistability whenever $\lambda_c^{-1}<\mu<\lambda_c$, where $\mu$ is the OR-to-AND rate ratio and $\lambda_c\simeq1.649$. The construction uses blocks of height $H$, width $H\lceil\log H\rceil$, and duration $O(H\log H)$. New cross-gated processes are also proved bistable in two dimensions for arbitrary positive competing rates, and in three dimensions at equal rates, under sufficiently weak, arbitrarily biased noise. The criterion is sufficient rather than a classification and excludes some drifting eroders lacking balanced stable directions. Results concern infinite-volume classical phase stability; exponential finite-size memory lifetimes and associated spectral-gap behavior are not established.